K -theory of endomorphisms, the TR -trace, and zeta functions
Jonathan A. Campbell, John A. Lind, Cary Malkiewich, Kate Ponto, Inna Zakharevich
aa r X i v : . [ m a t h . A T ] J un K -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETAFUNCTIONS JONATHAN A. CAMPBELL, JOHN A. LIND, CARY MALKIEWICH, KATE PONTO,AND INNA ZAKHAREVICHA
BSTRACT . We show that the characteristic polynomial and the Lefschetz zeta functionare manifestations of the trace map from the K -theory of endomorphisms to topologicalrestriction homology (TR). Along the way we generalize Lindenstrauss and McCarthy’smap from K -theory of endomorphisms to topological restriction homology, defining it forany Waldhausen category with a compatible enrichment in orthogonal spectra. In partic-ular, this extends their construction from rings to ring spectra. We also give a revisionisttreatment of the original Dennis trace map from K -theory to topological Hochschild ho-mology (THH) and explain its connection to traces in bicategories with shadow (alsoknown as trace theories). C ONTENTS
1. Introduction 12. Preliminaries: duality, bicategories, and spectra 63. Spectral categories and spectral Waldhausen categories 124. Bimodules over spectral categories and their traces 155. The additivity theorem for THH, revisited 196. The Dennis trace 247. The equivariant Dennis trace 308. The trace to topological restriction homology 369. Characteristic polynomials, zeta functions, and the Reidemeister trace 41Appendix A. Model categories of restriction systems 51References 531. I
NTRODUCTION
The trace of a matrix is one of the most fundamental invariants in mathematics.It is concrete, computable, easy to define, and ubiquitous. It generalizes to traces ofoperators, traces of endomorphisms of projective modules, traces in symmetric monoidalcategories [DP80], and traces in bicategories with shadow [Pon10, PS13, Kal15]. Thetrace is computable because it is additive: given two endomorphisms of k -vector spaces f : V → V and g : W → W , the trace satisfiestr( f ⊕ g ) = tr( f ) + tr( g ).A similar additivity statement holds for exact sequences of R -modules, in symmetricmonoidal categories [May01], and in bicategories [PS18]. Mathematics Subject Classification.
Therefore the trace, considered as a function from the set of matrices to the groundring, can be encoded using a universal additive invariant. The Hattori–Stallings trace K ( A ) −→ HH ( A ) ∼= A /[ A , A ],and its generalization the Dennis trace K ( A ) → HH( A ), make this idea precise. Here K ( A ) is the algebraic K -theory of a ring A [Qui73, Wal85] and HH is the Hochschildhomology. Following the outline of Goodwillie [Goo91], the Dennis trace was furthergeneralized to a map to topological Hochschild homology THH( A ), then to topologicalrestriction homology TR( A ) and topological cyclic homology TC( A ) in the celebrated workof Bökstedt, Hsiang, and Madsen [Bök85, BHM93]. The invariants THH, TR and TC arethe source of much of our computational knowledge of algebraic K -theory.The Hattori-Stallings trace is constructed in a concrete way from the ordinary traceof endomorphisms of modules. In this paper we show that the same is true of the Dennistrace and its refinements to THH and TR: they also encode concrete and computabletrace invariants. This is a shift in perspective, because typically THH, TR, and TC areviewed as tools for computing the whole of K -theory, rather than a sequence of naturalreceptacles for trace maps. Our goals are two-fold: • To explain why the invariants comprising the Dennis trace K ( A ) → THH( A ) andthe TR trace K ( A ) → TR( A ) are generalized traces arising in the bicategorical du-ality theory of Ponto and Ponto-Shulman [Pon10, PS13]. These invariants, whichinclude the trace of a matrix, the characteristic polynomial, and the Lefschetzzeta function, are easy to define, frequently computable, and have excellent for-mal properties. • To carefully explicate the construction of the Dennis trace map and its gen-eralizations. We follow previous accounts of the Dennis trace [DM96, BM12,DGM13], using shadows in bicategories to simplify and conceptualize the defini-tion.As a result of the first goal, we also show that fixed-point and periodic-point invariantsof “Reidemeister type” lift along the Dennis trace, as in [Iwa99, GN99].In summary, we view THH not as a stepping stone to K -theory computations, but asan important receptacle for invariants in its own right. This shift in perspective is ac-companied by a shift in emphasis in the definition of the Dennis trace. Cyclic invariancehas been central to the construction of the Dennis trace since its invention by Dennis[Wal79, p.36]. In that guise, cyclicity is more commonly called the Dennis–Waldhausen–Morita argument [BM11]. We expand this idea, putting it in the context of bicategoricaltraces.1.1. Statement of results: Invariants.
In order to relate the Dennis trace to bicate-gorical traces, we consider a generalization of the Dennis trace of the form e K ( A ; M ) −→ TR( A ; M ) −→ THH( A ; M )which was studied by Lindenstrauss and McCarthy [LM12] in the case of discrete ringsand bimodules. Here THH( A ; M ) denotes topological Hochschild homology with coeffi-cients in an ( A , A )-bimodule M , K ( A ; M ) is the K -theory of perfect A -modules P andtwisted endomorphisms(1.1) f : P → M ⊗ A P ,and e K ( A ; M ) is the cofiber of the map K ( A ) → K ( A ; M ) that sends each perfect A -moduleto its zero endomorphism. -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 3 We will recall in §2 that a twisted endomorphism f : P −→ M ⊗ A P , with P a dualizable A -module, has an associated bicategorical trace (Definition 2.14)tr( f ) : S −→ THH( A ; M )Our first result, an elaboration of [CP19, 7.11], says that the Dennis trace encodes thebicategorical trace. Theorem 1.2 (Example 6.26) . For any ring or ring spectrum A and a ( A , A ) -bimoduleM, there is a generalized Dennis trace map (Definition 6.16) e K ( A ; M ) trc / / THH( A ; M ) that on π takes the class of an endomorphism f : P → M ⊗ A P to its bicategorical trace tr( f ) : S → THH( A ; M ) . More generally, topological restriction homology encodes the traces of the iterates ofan endomorphism.
Theorem 1.3 (Theorem 8.21) . There is a lift of the Dennis trace to topological restrictionhomology (Definition 8.13) e K ( A ; M ) trc / / TR( A ; M ) that on π takes the class of an endomorphism f : P → M ⊗ A P to the trace of its n-folditerate f ◦ n : P → M ⊗ A · · · ⊗ A M ⊗ A Pfor every n ≥ . We call the map in Theorem 1.3 the TR -trace . The characteristic polynomial of amatrix is a refinement of the trace, and is encoded by the TR-trace.
Theorem 1.4 (Theorem 9.9) . If A is a discrete commutative ring, then the composite e K ( A ; A ) trc / / π TR( A ) ∼= / / (1 + tA [[ t ]]) × takes the class [ f : P → P ] of an endomorphism to its characteristic polynomial det(1 − t f ) . We emphasize that Theorem 1.4 states that the TR-trace is exactly the homotopicalanalogue of the characteristic polynomial. Since zeta functions are built out of charac-teristic polynomials, we summarize with the slogan:
K -theory is the natural home for additive invariants,
THH is the naturalhome for traces, and TR is the natural home for zeta functions. A related slogan occurs in topological fixed-point theory:THH is the natural home for fixed-point invariants and TR is the naturalhome for periodic-point invariants. The following result captures this idea, and is the topological analogue of the algebraicslogan.
Theorem 1.5 (Theorems 9.22 and 9.33) . Every self-map f : X → X of a connected finitecomplex defines a canonical class in endomorphism K -theory [ f ] ∈ K ( S [ Ω X ]; S [ Ω f X ]). The image of this class under the TR -trace coincides with the periodic-point invariantR ( Ψ ∞ ( f )) studied in [MP18b] .Composing with the map on TR induced by the ring map S [ Ω X ] collapse −−−−−→ S unit −−−→ H Z , J. A. CAMPBELL, J. A. LIND, C. MALKIEWICH, K. PONTO, AND I. ZAKHAREVICH the image in π TR( Z ) ∼= (1 + t Z [[ t ]]) × is the Lefschetz zeta function of f : ζ ( t ) = exp à ∞ X n = L ( f ◦ n ) t n n ! .In more detail, the image of [ f ] in π TR( S [ Ω X ]; S [ Ω f X ]) is given by the Fuller traces R ( Ψ n f ) C n for all n ≥
1. These are the strongest invariants that detect the n -periodicpoints of f up to homotopy, and our work here extends [MP18b] by lifting them to the K -theory of spherical group rings. This realizes a vision of Klein, McCarthy, Williamsand others about the fundamental nature of these periodic-point invariants.The theorem also suggests that the higher homotopy groups of K -theory with coeffi-cients capture parameterized versions of the Lefschetz zeta function, just as K -theorywithout coefficients captures parametrized Euler characteristics [DWW03]. We intendto return to this idea in future work.1.2. Statement of Results: The Dennis Trace.
In order to prove that the trace mapsout of K -theory encode bicategorical traces, as described in the theorems above, we in-tegrate the perspective of shadows into the construction of the Dennis trace. This hasthe unexpected benefit of simplifying many aspects of its construction. We emphasizethat our definition is similar to and very much motivated by the work in [DM96, BM12,DGM13], but the focus on shadows is conceptually clarifying.To make sense of both the algebraic K-theory of a category and its topological Hochschildhomology we need the category to be a spectral category and have a compatible Wald-hausen structure. Applying the building blocks of algebraic K -theory (i.e. applying w • and S • ) to a Waldhausen category goes back to Waldhausen’s original work, but applyingthese to a spectral category is the most technically demanding portion of the paper. Forthe introduction we will treat this step as a black box.Given a spectral category C and a Waldhausen category C with appropriate compat-ibility (Definition 3.9), the foundation of the Dennis trace is the inclusion of the zeroskeleton in THH: _ f ∈ End( C ) S → THH( C ).Note that the object on the left depends only on the base category C , which we assumedto be Waldhausen. Since w • and S • can be applied to both C and C , the inclusion of thezero skeleton gives a map of bisimplicial spectra Σ ∞ ob w • S • End( C ) → THH( w • S • C )and more generally for each n ≥ n + Σ ∞ ob w • S ( n ) • ,..., • End( C ) → THH( w • S ( n ) • ,..., • C ).The Dennis trace is then defined to be a map in the homotopy category(1.6) trc : K (End( C )) −→ THH( C )obtained from a zig-zag of the form Σ ∞ ob w • S ∗• ,..., • End( C ) → THH( w • S ∗• ,..., • C ) ≃ ←− THH( S ∗• ,..., • ) ≃ ←− Σ ∞ THH( C ).The backwards maps of the zig-zag are provided by the following two theorems. Theorem 1.7 (Lemma 3.15) . If C is a spectral category and w k C is the associated categoryof flags of weak equivalences in C , then there is a natural equivalence THH( w k C ) ≃ ←− THH( C ). -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 5 Theorem 1.8 (Additivity of THH, Theorem 5.1) . Let C be a spectral category and letS C be the associated spectral category of cofiber sequences in C . Then there is a naturalequivalence THH( S C ) ≃ ←− THH( C ) ∨ THH( C ). These equivalences inductively define an equivalence
THH( S • C ) ≃ ←− Σ THH( C ), and thus an equivalence to the iterated S • -construction THH( S ( n ) • ,..., • C ) ≃ ←− Σ n THH( C ).Note that, as a result, the zig-zag defining (1.6) has two spectral directions. Onespectral direction comes from the enrichment of C . The other spectral direction comesfrom the iterated S • -construction and additivity.The above two theorems are essential components in the construction of the Dennistrace. They are well known in many different contexts [DM96, BM11, BM12, DGM13,HS18]. We provide new proofs in the context of spectral Waldhausen categories thathighlight how these theorems are completely formal consequences of the fact that • THH is a shadow on the bicategory of spectral categories and spectral bimodules,and that • THH preserves cofiber sequences in the bimodule slot.We define the Dennis trace for any ring or ring spectrum A by applying the above tothe spectral Waldhausen category P A of perfect A -module spectra:(1.9) K (End( A )) : = K (End( P A )) −→ THH( P A ).To make this land in THH( A ) we use one final core result, which is also a formal conse-quence of the shadow property. Theorem 1.10 (Morita invariance of THH, Example 4.16) . There is a natural equiva-lence
THH( P A ) ≃ −→ THH( A ) defined by a bicategorical trace. Again, this is well known, but recognizing that the map underlying the equivalence isitself a bicategorical trace is clarifying and simplifies the proof.The trace to THH( A ; M ) for an ( A , A )-bimodule M proceeds in the same way, usingvariants of the above theorems with coefficients. To define the lift to TR as in [LM12] weperform the same manipulations but replace the endomorphisms c → c in C by length r cycles of maps(1.11) a f −→ a f −→ a f −→ · · · f r − −−→ a r f f −→ a for each r ≥
1, and include these into the zero skeleton of THH ( r ) ( C ), a certain r -foldsubdivision of THH. The resulting traces agree by taking fixed points along the actionof a cyclic group that rotates the endomorphisms, and therefore they assemble togetherinto a map to TR.1.3. Connection to the literature.
In the case of discrete or simplicial rings A , thetrace of Theorem 1.2 is not new. The algebraic K -theory of parametrized endomorphisms K ( A ; M ) and its trace to THH( A ; M ) were first defined in [DM94] for exact categories, seealso [Iwa99, DGM13]. The lift to TR( A ; M ) was constructed for discrete rings (or exactcategories) by Lindenstrauss and McCarthy in [LM12]. Our contribution is mainly tore-tool the construction so that it works for any ring spectrum, or more generally anyspectrally enriched Waldhausen category. J. A. CAMPBELL, J. A. LIND, C. MALKIEWICH, K. PONTO, AND I. ZAKHAREVICH
Our reworking uses the Hill-Hopkins-Ravenel equivariant norm of [HHR16] and theassociated cyclotomic structure on THH from [ABG +
18, Mal17a, DMP + π we make heavy use of the mainresult of [CP19].In the setting of stable ∞ -categories, the Dennis trace has a universal characteri-zation [BGT13, BGT16]. The point-set model of the Dennis trace for spectrally en-riched Waldhausen categories serves as a concrete description of the trace for stable ∞ -categories. (Note from [BGT13] that the two settings are essentially equivalent.) Weexpect that the generalized Dennis trace constructed here will similarly underlie the ∞ -categorical Dennis trace with coefficients [BGT16, HS18], and the universal charac-terization of the TR-trace described in forthcoming work of Nikolaus [Nik].Finally, on the subject of fixed-point theory, we note that Theorem 1.5 is closely relatedto the main result of [Iwa99], which lifts the Reidemeister traces of the iterates R ( f ◦ n )to K ( Z [ π X ]; Z [ π X f ]). They are related because on π , the Fuller trace R ( Ψ n f ) C n isequivalent to the Reidemeister traces R ( f ◦ k ) for all k | n , by [MP18b]. We anticipate thatthe formulation in Theorem 1.5 will be needed for future generalizations to familiesof endomorphisms, where the Fuller trace becomes a strictly stronger invariant than R ( f ◦ n ), and approaches that use discrete rings tend to break down.1.4. Organization.
We recall preliminaries on duality and traces in symmetric mon-oidal categories and bicategories, as well as on equivariant spectra, in §2. §3–4 recalland extend necessary foundations to apply the trace in categories that are compatiblyspectrally enriched and have a Waldhausen structure. In §5 we revisit the additivityof THH using shadows in preparation for the definition of the Dennis trace in §6. Weextend this definition to an equivariant trace in §7 and use it to define the TR trace in§8. Finally in §9 we describe applications to homotopical characteristic polynomials andperiodic point invariants.1.5.
Acknowledgments.
JC would like to thank Andrew Blumberg, Mike Mandell,and Randy McCarthy for helpful conversations about this paper, and for general wis-dom about trace methods. CM would like to thank Randy McCarthy for persistentlytelling him about the TR trace for years – it’s beginning to sink in a little. KP was par-tially supported by NSF grant DMS-1810779 and the University of Kentucky RoysterResearch Professorship. The authors thank Cornell University for hosting the initialmeeting which led to this work.2. P
RELIMINARIES : DUALITY , BICATEGORIES , AND SPECTRA
We begin with a slogan:
Every endomorphism of a finite mathematical object defines a class inK -theory, and the Dennis trace takes its trace.
In this section, we recall many of the fundamental definitions in this slogan. We de-fine the trace of an endomorphism in a symmetric monoidal category, and then extendthe formalism to the noncommutative setting of bicategories. Ideas suggesting thisapproach can be found in [Nic05], but the first successful formalization was the no-tion of “shadowed bicategory” in the thesis of the fourth author [Pon10, PS13], laterre-discovered by Kaledin under the name “trace theory” [Kal15, Kal20]. In this sec-tion we give a brief introduction to these ideas. The reader is encouraged to consult[Pon10, PS13, PS14, DP80] for more details. -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 7 Duality and trace in symmetric monoidal categories.
An object X of a sym-metric monoidal category ( C , ⊗ , I ) is dualizable if there exists an object X ∗ , togetherwith an evaluation map ǫ : X ⊗ X ∗ → I and a coevaluation map η : I → X ∗ ⊗ X , such thatboth composites X X ⊗ I X ⊗ X ∗ ⊗ X I ⊗ X XX ∗ I ⊗ X ∗ X ∗ ⊗ X ⊗ X ∗ X ∗ ⊗ I X ∗∼= id ⊗ η ǫ ⊗ id ∼=∼= η ⊗ id id ⊗ ǫ ∼= are identity maps. The dual object X ∗ is unique up to canonical isomorphism.Given a dualizable object X , the trace of a map f : X → X is the composite(2.1) tr( f ) : I X ∗ ⊗ X X ∗ ⊗ X X ⊗ X ∗ I . η id ⊗ f ∼= ǫ When f is the identity morphism, we call tr(id X ) the Euler characteristic of the object X [DP80, LMSM86, PS14]. Example 2.2.
In classical contexts, the above definition becomes familiar. i . In the category of vector spaces over a field k , the trace of an endomorphism f : V −→ V of a finite dimensional vector space is the k -linear map tr( f ) : k −→ k given by multiplication by the trace of a matrix representing f . ii . In the stable homotopy category of spectra, the trace of the identity map on thesuspension spectrum Σ ∞+ X of a finite CW complex X is a map tr(id Σ ∞+ X ) : S → S whose degree is the Euler characteristic of X . iii . More generally, if f : X −→ X is a self-map of a finite CW complex, then thetrace of the stable map Σ ∞+ f : Σ ∞+ X −→ Σ ∞+ X is the Lefschetz number L ( f ) [DP80,Dol65].2.2. Bicategories and shadows. If A is a non-commutative ring then A -modules donot form a symmetric monoidal category. Hence the trace as defined in (2.1) does notmake sense. To take the trace of an endomorphism f : M → M in this setting, one mustcircumvent the problem that M ⊗ N ≇ N ⊗ M — they are not even objects of the same type. The Hattori–Stallings trace solves this is-sue in an ad-hoc way, by modding out by a commutator ideal. The general solution to thisissue first appeared in [Pon10] (and was independently developed in work of Kaledin oncyclic K -theory [Kal15]). The idea is to use bicategories to encode noncommutativity, andcreate a type of wrapper 〈〈−〉〉 , called a “shadow,” which removes just enough informationto give us commutativity when we need it. Definition 2.3. A bicategory B consists of objects, A , B , . . ., called 0-cells, and categor-ies B ( A , B ) for each pair of objects A , B . Objects in the category B ( A , B ) are called 1-cellsand morphisms are called 2-cells. The bicategory is further equipped with horizontalcomposition functors(2.4) ⊙ : B ( A , B ) × B ( B , C ) → B ( A , C ),that are associative and have units U A ∈ B ( A , A ), up to coherent isomorphism.In our context, the horizontal composition will substitute for the tensor product; thefollowing family of examples is used throughout this section as motivation. Example 2.5.
There is a bicategory with one 0-cell for each ring A . For each pair ofrings A and B , the category B ( A , B ) is the category of ( A , B )-bimodules. The horizontalcomposition is the tensor product ⊗ B . J. A. CAMPBELL, J. A. LIND, C. MALKIEWICH, K. PONTO, AND I. ZAKHAREVICH
This bicategory serves as motivation for the bicategory of spectral categories, bimod-ules, and homotopy classes of maps of bimodules, which we describe in §4. The true workof the paper requires the bicategory in §4.In order to define the trace, extra structure is required.
Definition 2.6 ([Pon10]) . Let B be a bicategory. A shadow functor for B consists ofthe following data: a target category: T , functors: 〈〈−〉〉 : B ( C , C ) → T for each object C of B , a natural isomorphism: (2.7) θ : 〈〈 M ⊙ N 〉〉 ∼= −→ 〈〈 N ⊙ M 〉〉 for M ∈ B ( C , D ) and N ∈ B ( D , C ).These must satisfy the condition that the following diagrams commute whenever theymake sense: cyclic associativity: 〈〈 ( M ⊙ N ) ⊙ P 〉〉 θ / / 〈〈 a 〉〉 (cid:15) (cid:15) 〈〈 P ⊙ ( M ⊙ N ) 〉〉 〈〈 a 〉〉 / / 〈〈 ( P ⊙ M ) ⊙ N 〉〉〈〈 M ⊙ ( N ⊙ P ) 〉〉 θ / / 〈〈 ( N ⊙ P ) ⊙ M 〉〉 〈〈 a 〉〉 / / 〈〈 N ⊙ ( P ⊙ M ) 〉〉 θ O O . unitality: 〈〈 M ⊙ U C 〉〉 θ / / 〈〈 r 〉〉 & & ▼▼▼▼▼▼▼▼▼▼▼ 〈〈 U C ⊙ M 〉〉〈〈 l 〉〉 (cid:15) (cid:15) θ / / 〈〈 M ⊙ U C 〉〉〈〈 r 〉〉 x x qqqqqqqqqqq 〈〈 M 〉〉 .If 〈〈−〉〉 is a shadow functor on B , then the composite 〈〈 M ⊙ N 〉〉 θ / / 〈〈 N ⊙ M 〉〉 θ / / 〈〈 M ⊙ N 〉〉 is the identity [PS13, Prop. 4.3]. More generally, the circular product 〈〈 M ⊙ · · · ⊙ M n 〉〉 ofany composable list of 1-cells M , . . ., M n is well-defined up to canonical isomorphism[MP18a, 1.6]. Example 2.8.
The 0th Hochschild homology group 〈〈 M 〉〉= HH ( A ; M ) : = M /( am − ma )defines a shadow on the bicategory of rings and bimodules. The isomorphism θ : HH ( A , M ⊗ B N ) → HH ( B , N ⊗ A M )is given by observing that both sides are the same quotient of M ⊗ N .If we modify the bicategory of Example 2.5 by taking derived tensor products ⊗ L instead of ordinary ones, then the higher Hochschild homology HH ∗ ( A ; M ) is also ashadow. See Definition 4.9 for an analog of this using topological Hochschild homology.While we won’t make any formal use of graphical reasoning or string diagram calculi,“cartoon” images of the shadow and later generalizations can be useful. Figure 2.9 con-tains two examples of this. We think of a 1-cell M as a vertex with two edges labeledby the 0-cells which are the source and target of M . Then the shadow of M glues thefree ends of these edges to each other, as in Figure 2.9a. The shadow of the horizontalcomposite of compatible 1-cells is displayed in Figure 2.9b. -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 9 AM ( A ) 〈〈 M 〉〉 A BMN ( B ) 〈〈 M ⊙ N 〉〉 F IGURE
A AQ A BAM NQ A BBM PN BB P η ⊙ id id ⊙ f id ⊙ ǫ F IGURE
Duality and trace.
With a shadow we can now define traces in bicategories. Westart by recalling the generalization of dualizability to bicategories.
Definition 2.10.
We say that a 1-cell P ∈ B ( C , D ) in a bicategory is left dualizable ifthere is a 1-cell P ∗ ∈ B ( D , C ), called its left dual , and coevaluation and evaluation 2-cells η : U D → P ∗ ⊙ P and ǫ : P ⊙ P ∗ → U C satisfying the triangle identities. We say that( P ∗ , P ) is a dual pair , that P ∗ is right dualizable , and that P is its right dual . Example 2.11. i . For rings C and D , a ( C , D )-bimodule P is left dualizable if and only if it is finitelygenerated and projective as a left C -module. ii . The 2-category of small categories, functors, and natural transformations is abicategory. The functors and their compositions may either be written from rightto left (function convention), or from left to right (bimodule convention, (2.4)).Under the function convention, a functor G : C → D is left dualizable if andonly if it is a left adjoint. Under the bimodule convention, G is left dualizable ifand only if it is a right adjoint. Definition 2.12 ([Pon10]) . Let B be a bicategory with a shadow functor and let ( P ∗ , P )be a dual pair of 1-cells. Let M ∈ B ( C , C ) and N ∈ B ( D , D ) be 1-cells. The trace of a 2-cell f : P ⊙ N → M ⊙ P is the composite 〈〈 N 〉〉∼= 〈〈 U D ⊙ N 〉〉 〈〈 η ⊙ id N 〉〉−−−−−→ 〈〈 P ∗ ⊙ P ⊙ N 〉〉 〈〈 id P ∗ ⊙ f 〉〉−−−−−−→ 〈〈 P ∗ ⊙ M ⊙ P 〉〉 θ −→ 〈〈 M ⊙ P ⊙ P ∗ 〉〉 〈〈 id M ⊙ ǫ 〉〉−−−−−→ 〈〈 M ⊙ U C 〉〉∼= 〈〈 M 〉〉 . The trace for a 2-cell g : N ⊙ P ∗ → P ∗ ⊙ M is defined similarly.See Figure 2.13.As explained in [Pon10], there is a conceptual re-interpretation of the Hattori–Stallingstrace of an A -module endomorphism f : P → P as a bicategorical trace Z ∼= HH ( Z ) HH ( Z ; P ∗ ⊗ A P ) HH ( Z ; P ∗ ⊗ A P ) A /[ A , A ].HH ( A ; P ⊗ Z P ∗ ) HH ( A ) η f ∼= ǫ ∼= This formalism is precisely what we need to generalize the classical link between theDennis trace and the Hattori–Stallings trace so that it also applies to ring spectra.
Definition 2.14. If P ∈ B ( C , D ) and P is left dualizable with left dual P ∗ , the Eulercharacteristic of P is the trace of its identity 2-cell, χ ( P ) : 〈〈 U D 〉〉→ 〈〈 U C 〉〉 χ ( P ) : = tr(id P ).(Here, M = U C and N = U D .) Similarly, the Euler characteristic χ ( P ∗ ) is the trace of theidentity 2-cell of P ∗ . A check of the definitions shows that χ ( P ) = χ ( P ∗ ).As for symmetric monoidal categories, there is also a notion of invertible 1-cell that isstronger than being dualizable. It gives us a natural notion of equivalence between the0-cells. Definition 2.15 ([Bén67][CP19, Def. 4.1]) . A pair of 1-cells P ∈ B ( C , D ) and P ∗ ∈ B ( D , C )forms a Morita equivalence between C and D if ( P ∗ , P ) is a dual pair whose coevalua-tion and evaluation maps are isomorphisms. Example 2.16.
Morita equivalence in the bicategory of rings and bimodules is the usualnotion of Morita equivalence between rings.When ( P ∗ , P ) is a Morita equivalence, the Euler characteristic χ ( P ) is an isomor-phism since it is a composite of isomorphisms. We will make significant use of thisobservation—it is an essential part of our approach to Theorem 1.10.We finish this section with a definition which will be used often in this paper: Definition 2.17.
Let C be a category, such as the category of rings and ring homo-morphisms. A pre-twisting of an object C ∈ C is a pair of morphisms F : A → C and G : B → C ; this is denoted A / C / B F G . When clear from context we often omit A and B from the notation. A morphism of pre-twistings ( H , I , J ) : A / C / B F G → A ′ / C ′ / B F ′ G ′ isa commutative diagram A F / / H (cid:15) (cid:15) C J (cid:15) (cid:15) B G o o I (cid:15) (cid:15) A ′ F ′ / / C ′ B ′ G ′ o o A pre-twisting is a twisting if A = B ; we denote a twisting by A / C F G , and often omit A from the notation. If A / C F G → A ′ / C ′ F ′ G ′ is a morphism of pre-twistings between twist-ings then it is a morphism of twistings if H = I ; this is denoted ( I , J ).Many bimodules of interest arise from twistings in the following way. Example 2.18.
Let R be a ring, and let f : S → R and g : T → R be a pre-twisting of R . Then S / R / T f g gives R the structure of an ( S , T )-bimodule, with S acting on theleft through f and T acting on the right through g . Morphisms of twistings producemorphisms of bimodules. In a similar manner, a twisting S / R f g gives R an S -bimodulestructure.We use this perspective in future sections (e.g. Definition 4.4) to produce examples ofbimodules over spectral categories.2.4. Review of orthogonal G -spectra. We will also recall a bit of the theory of equi-variant spectra; more details can be found in [MM02, HHR16, CLM + ].For simplicity, let G be a finite abelian group, such as C r = Z / r Z . An orthogonal G -spectrum is an orthogonal spectrum with an action of G . By the point-set changeof universe functor, this is the same thing as an orthogonal spectrum indexed on thefinite-dimensional representations of G [MM02, V.1.5]. An equivalence of G -spectra is a map that induces an isomorphism on the homotopy groups that are defined using -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 11 all of the G -representations; these are the weak equivalences in a model structure onorthogonal G -spectra from [MM02, III.4.2], and when we say “cofibrant” or “fibrant” weare always referring to the notions coming from this model structure.There is a categorical fixed-points functor ( − ) H from orthogonal G -spectra to orthog-onal G / H -spectra. It is right Quillen and its right-derived functor is the genuine fixedpoints functor. There is also a point-set geometric fixed points functor Φ H from orthog-onal G -spectra to orthogonal G / H -spectra [MM02, V.4.1]. It is not a left adjoint, but it isleft-deformable and we refer to its left-derived functor L Φ H as the (left-derived) geo-metric fixed points functor. On suspension spectra we have canonical isomorphisms Φ H Σ ∞ X ∼= Σ ∞ X H . There is a natural transformation κ : X H → Φ H X for G -spectra X called the restriction map . Making X cofibrant and fibrant gives a corresponding re-striction map from the genuine fixed points to the geometric fixed points.We recall a common tool for checking that a map of G -spectra is an equivalence. Proposition 2.19. [May96, XVI.6.4]
A map of G-spectra X → Y is an equivalence if andonly if for every H ≤ G the induced map on derived geometric fixed points L Φ H X → L Φ H Yis an equivalence of spectra.
As discussed in [DMP +
19, 4.1], the geometric fixed point functor Φ H and its left-derived functor L Φ H also commute with the forgetful functor from G -spectra to H -spectra. As a result, to determine whether a map of G -spectra is an equivalence, itsuffices to forget down to the H -action and measure its geometric H -fixed points, foreach H ≤ G .If X is an orthogonal spectrum, then the r -fold smash product X ∧ r admits a canonical C r -action by rotating the factors. By the above discussion, we may then consider X ∧ r tobe an orthogonal C r -spectrum. This is the Hill–Hopkins–Ravenel norm of X . The fol-lowing fundamental property of the norm gives us control over its equivariant homotopytype. Proposition 2.20. [HHR16]
There is a natural diagonal map of orthogonal spectraD r : X −→ Φ C r X ∧ r . If X is cofibrant, then X ∧ r is cofibrant and D r is an isomorphism on the point-set level.We therefore get a natural equivalence for cofibrant X ,X ≃ L Φ C r X ∧ r .In [ABG +
18, Mal17a] this result is used to build a cyclotomic structure on the topologi-cal Hochschild homology of an orthogonal ring spectrum, that by [DMP +
19] is equivalentto the cyclotomic structure of Bökstedt [Bök85]. In this paper we use Proposition 2.20in much the same way to control the equivariant homotopy type of the r -fold topologicalHochschild homology spectrum THH ( r ) (see Definition 7.4). Remark . It is especially important for us to note that on suspension spectra, theHHR norm agrees with the more obvious map Σ ∞ X ∼= Σ ∞ ( X ∧ r ) C r ∼= Φ C r Σ ∞ X ∧ r ∼= Φ C r ( Σ ∞ X ) ∧ r .This can be checked by tracing through the definitions, but it is much easier to concludeit formally by noting that any point-set automorphism of the functor Σ ∞ X has to be theidentity when X = S and therefore has to be the identity for all X . The rigidity theoremfor geometric fixed points from [Mal17a, 1.2] is a generalization of this observation.
3. S
PECTRAL CATEGORIES AND SPECTRAL W ALDHAUSEN CATEGORIES
In this section we establish our conventions on spectral categories, define the notionof a spectral Waldhausen category, and set up notation. In later sections, spectral cate-gories will play the role that rings played in the primary example of §2.3.1.
Spectral categories.Definition 3.1. A spectral category C is a category enriched in orthogonal spectra. Inother words, for every ordered pair of objects ( a , b ) there is a mapping spectrum C ( a , b ),a unit map S −→ C ( a , a ) from the sphere spectrum for every object a , and multiplicationmaps C ( a , b ) ∧ C ( b , c ) −→ C ( a , c )that are strictly associative and unital. A spectral category is pointwise cofibrant ifall mapping spectra are cofibrant in the stable model structure on orthogonal spectra[MMSS01, §9].A functor of spectral categories F : C −→ D consists of a map on the object setsand maps of spectra F : C ( a , b ) −→ D ( F a , F b ) that agree with the multiplications andunits. Such a functor is called a
Dwyer–Kan embedding if each of these maps is anequivalence [BM12, 5.1].Throughout, we assume that spectral categories are small, meaning that they have aset of objects.
Remark . Our convention that C ( a , b ) is an orthogonal spectrum imposes no essentialrestriction. Any category enriched in symmetric or EKMM spectra can be turned into anorthogonal spectral category using the symmetric monoidal Quillen equivalences ( P , U )and ( N , N ) from [MMSS01] and [MM02], respectively. Example 3.3. i . Every (orthogonal) ring spectrum A is a spectral category with one object. ii . If C is a pointed category, then there is a spectral category Σ ∞ C with the sameobjects as C , mapping spectra given by the suspension spectra Σ ∞ C ( a , b ), andcomposition arising from C . Definition 3.4. A base category of a spectral category C is a pair ( C , F : Σ ∞ C → C )where C is an pointed category and F is a spectral functor that is the identity on objectsets. When the functor is clear from context we omit it from the notation.We can form such a base category C by restricting each mapping spectrum to levelzero and forgetting the topology. However, there are also examples, such as Example 3.7,which do not arise in this way. Definition 3.5.
Let A be an orthogonal ring spectrum. The category of A -modules M A is a spectral category whose objects are the cofibrant module spectra over the ringspectrum A . The mapping spectra are the right-derived mapping spectra. When A isclear from context we omit it from the notation.Perfect modules also form a spectral category. Example 3.6.
For a ring spectrum A , the category of perfect A -modules P A is the fullsubcategory of M A spanned by the modules that are retracts in the homotopy categoryof finite cell A -modules. When A is understood, we call this P . There is a functor ofspectral categories A → P A taking A to A equipped with the left-multiplication action.In both of these cases, we make the mapping spectra right-derived by passing throughthe category of EKMM spectra, using the symmetric monoidal Quillen adjunction ( N , N )from [MM02, I.1.1]. A more explicit treatment appears in [CLM + , Section 3.2].Another common example of a spectral category is a functor category. -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 13 Example 3.7.
Let I be a small category, and let C be a spectral category with a base cat-egory C . Write Fun( I , C ) for the category of functors (and natural transformations) I → C . The category Fun( I , C ) is a base category of a spectral category Fun( I , C ), whosemapping spectra are right-derived from the equalizereq ³ Y i ∈ ob I C ( φ ( i ), γ ( i )) ⇒ Y i −→ i C ( φ ( i ), γ ( i )) ´ .To be more precise, the spectral category Fun( I , C ) is defined using the Moore end con-struction of [MS02, 2.4] and [BM11, 2.3]. An explicit and detailed treatment of thisconstruction for spectra appears in [CLM + , Section 4].Many of our techniques will require the use of pointwise cofibrant spectral categories.Spectral categories can always be replaced with equivalent pointwise cofibrant spectralcategories using the model structure from [SS03, 6.1, 6.3]. Theorem 3.8.
There is a pointwise cofibrant replacement functor Q and a pointwisefibrant replacement functor R on spectral categories. In particular,Q : SpCat → SpCat is a functor equipped with a natural transformation q : Q ⇒ id SpCat such that q C is apointwise equivalence for every spectral category C . Spectral Waldhausen categories.
We now extend the definition of Waldhausencategories to spectral categories. Recall that a Waldhausen category C is a categorywith cofibrations and weak equivalences satisfying the axioms in [Wal85, §1.2]. Definition 3.9. A spectral Waldhausen category is a spectral category C togetherwith a base category C which is equipped with a Waldhausen category structure. Thisdata is subject to the following three conditions: i . The zero object of C is also a zero object for C . ii . Every weak equivalence c −→ c ′ in C induces stable equivalences C ( c ′ , d ) ∼ −→ C ( c , d ), C ( d , c ) ∼ −→ C ( d , c ′ ). iii . For every pushout square in C along a cofibration a (cid:15) (cid:15) (cid:31) (cid:127) / / b (cid:15) (cid:15) c (cid:31) (cid:127) / / d and object e , the resulting two squares of spectra C ( a , e ) O O o o C ( b , e ) O O C ( c , e ) o o C ( d , e ) C ( e , a ) (cid:15) (cid:15) / / C ( e , b ) (cid:15) (cid:15) C ( e , c ) / / C ( e , d )are homotopy pushout squares.A functor of spectral Waldhausen categories F : ( C , C ) −→ ( D , D ) is an exactfunctor F : C −→ D and a spectral functor F : C −→ D such that the diagram Σ ∞ C Σ ∞ F / / (cid:15) (cid:15) Σ ∞ D (cid:15) (cid:15) C F / / D commutes. When it is clear from context, we omit C from the notation and refer simplyto the spectral Waldhausen category C . Example 3.10.
The categories P A of perfect A -modules and M A of all A -modules areboth spectral Waldhausen categories. Example 3.11. If C is a simplicially enriched Waldhausen category in the sense of[BM11] then the spectral enrichment C Γ from [BM11, 2.2.1] is compatible with the Wald-hausen structure in our sense. The same is true for the non-connective enrichment C S from [BM11, 2.2.5] if C is enhanced simplicially enriched. Proposition 3.12 ([CLM + , Theorem 4.1]) . The category of functors construction fromExample 3.7 respects Waldhausen structures and defines a functor
Fun :
Cat op × SpWaldCat → SpWaldCat , by giving Fun( I , C ) the levelwise Waldhausen structure. By levelwise Waldhausen structure, we mean that a map of diagrams φ → γ is a cofi-bration (resp. weak equivalence) if φ ( i ) → γ ( i ) is a cofibration (resp. weak equivalence)for every object i of the indexing category I . In practice, this is usually more cofibrationsthan we need, but we can always restrict the class of cofibrations: Lemma 3.13. If ( C , C ) is a spectral Waldhausen category, and C ′ is a different Wald-hausen structure on C with the same weak equivalences and fewer cofibrations, then ( C , C ′ ) is also a spectral Waldhausen category. The S • construction and the K -theory of spectral Waldhausen categories. Let [ k ] = { < < · · · < k } denote the totally ordered set on k + C , the S • construction produces a simplicial category whose k thlevel is the full subcategory S k C ⊆ Fun([ k ] × [ k ], C )consisting of functors that vanish on all pairs ( i , j ) with i > j , and on the remaining pairs(in other words the category of arrows Arr[ k ]) form a sequence of cofibrations and theirquotients [Wal85, §1.3].We extend this definition to spectral Waldhausen categories C by defining S k C to bethe full subcategory of Fun([ k ] × [ k ], C ) on the objects that define S k C . We define the n -fold S • construction and the simplicial category w • of composable sequences of weakequivalences for spectral Waldhausen categories in a similar way; see [CLM + , Defini-tions 5.6 and 5.10] for more details. Lemma 3.14 ([CLM + , Definition 5.10]) . For every n ≥ , there is a functor from spectralWaldhausen categories to multisimplicial Waldhausen categoriesw • S ( n ) • : SpWaldCat −→ Fun( ∆ n + , SpWaldCat ) which on base categories takes C to w • S ( n ) • C , as defined by Waldhausen [Wal85] . For the next lemma, recall that a
Dwyer–Kan equivalence is a Dwyer–Kan em-bedding of spectral categories that induces an equivalence of ordinary categories afterapplying π to the mapping spectra. Lemma 3.15 ([CLM + , Lemma 5.7]) . The iterated degeneracy mapw S k ,..., k n C −→ w k S k ,..., k n C is a Dwyer–Kan equivalence of spectral categories. In particular, the spectral categoriesw k C are all canonically Dwyer–Kan equivalent to C . Definition 3.16.
We define the K -theory of a spectral Waldhausen category C to be the K -theory of the base category C . In other words, the n th level of the K -theory spectrum -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 15 is obtained from the spectral category w • S ( n ) • C by restricting to objects and taking thegeometric realization: K ( C ) n = ¯¯ ob w • S ( n ) • C ¯¯ .Note that, as usual, the K -theory spectrum is a symmetric spectrum with the sym-metric groups permuting the S • terms.4. B IMODULES OVER SPECTRAL CATEGORIES AND THEIR TRACES
In this section, we define the bicategory of spectral categories and bimodules, whichis the relevant generalization of the bicategory of rings and bimodules from §2. Thisbicategory can be equipped with a shadow via THH, and we use the notion of Moritaequivalence from Proposition 4.12 to construct examples of equivalences on THH.4.1.
Spectral bimodules.Definition 4.1. If C and D are spectral categories, a ( C , D ) -bimodule is a spectral func-tor X : C op ∧ D → Spto the spectral category of orthogonal spectra. More explicitly, a bimodule X consists ofan orthogonal spectrum X ( c , d ) for every ordered pair ( c , d ) ∈ ob C × ob D , along with a left action by C and a right action by DC ( a , c ) ∧ X ( c , d ) −→ X ( a , d ), X ( c , d ) ∧ D ( d , e ) −→ X ( c , e )satisfying evident unit and associativity conditions. A morphism of ( C , D )-bimodules X → Y is a collection of maps of orthogonal spectra X ( c , d ) → Y ( c , d ) commuting with the C and D actions. A pointwise equivalence of bimodules is a morphism which inducesweak equivalences of spectra X ( c , d ) ∼ −→ Y ( c , d ) for all objects c ∈ C and d ∈ D . We denotethe category of spectral ( C , D )-bimodules by M ( C , D ) . Example 4.2. If A and B are ring spectra, then a ( B , A )-bimodule is the same thing asa bimodule over the associated one-object spectral categories. Example 4.3.
Let C be a spectral category. Then C gives rise to a ( C , C )-bimodule definedby the enrichment functor Hom : C op ∧ C → Sp. By an abuse of notation we denote thisbimodule by C .We can further generalize this example by allowing different sources for the domainsand codomains of the mapping spectra. Definition 4.4.
Recall from Definition 2.17 that a pre-twisting of spectral categories A / C / B F G is a pair of spectral functors F : A → C and G : B → C . Given a pre-twisting, wedefine an ( A , B )-bimodule (which, by an abuse of notation, we denote by C F G ) by C F G ( a , b ) : = C ( F ( a ), G ( b )).We have the following special cases: • the ( A , C )-bimodule C F id , abbreviated by C F or C A (when F is clear from context). • the ( C , B )-bimodule C id G , abbreviated by C G or C B (when G is clear from context). • the ( C , C )-bimodule C id id , abbreviated by C . Note that this agrees with the use of C as a bimodule above. Example 4.5.
Let A and B be ring spectra and let M be a cofibrant ( B , A )-bimodule. Asdiscussed in [CLM + , Definition 3.8], M induces a spectral functor on the categories ofmodules M ∧ A − : M A → M B , and thus defines a ( M B , M A )-bimodule ( M B ) M ∧ A − . We willoften abbreviate this to M M if the rings are understood. By [CLM + , Lemma 3.15], thereis a natural equivalence of ( B , A )-bimodules M → M M given by the map M → M B ( B , M ) adjoint to the B -action on M . Note that the target is implicitly restricted to a ( B , A )-bimodule along the canonical inclusions of spectral categories A → M A and B → M B .The bar construction provides a canonical model for the derived coend of bimodulesover spectral categories; this is the appropriate analog in our context of the derivedtensor product of bimodules over rings. Definition 4.6.
Let X be a ( C , D )-bimodule and let Y be a ( D , E )-bimodule. Define the two-sided categorical bar construction B ( X , D , Y ) to be the ( C , E )-bimodule whosevalue at ( c , e ) is the geometric realization of the simplicial spectrum B • ( X , D , Y )( c , e )given by B n ( X , D , Y )( c , e ) = _ d ,..., d n X ( c , d ) ∧ D ( d , d ) ∧ · · · ∧ ( d n − , d n ) ∧ Y ( d n , e ).As is usual for bar constructions, the iterated D -action maps define canonical pointwiseequivalences(4.7) B ( D , D , Y )( d , e ) ∼ −→ Y ( d , e ) and B ( X , D , D )( c , d ) ∼ −→ X ( c , d ).When X is a ( C , C )-bimodule we define the topological Hochschild homology or cyclic bar construction THH( C ; X ) to be the realization of the simplicial spectrum B cy • ( C ; X ) given by B cy n ( C ; X ) : = _ c ,..., c n C ( c , c ) ∧ C ( c , c ) ∧ · · · ∧ C ( c n − , c n ) ∧ X ( c n , c ).When C is pointwise cofibrant, the definition is equivalent to all other definitions of THHin the literature, e.g. [Bök85, BM11, DGM13, NS18].Note that THH( C ; X ) is functorial in both C and X . Directly from the definition we getthe following observation: Lemma 4.8.
A morphism of twistings A / C F G → A ′ / C ′ F ′ G ′ induces a morphism THH( A ; C F G ) → THH( A ′ ; C ′ F ′ G ′ ).We equip the category M ( C , D ) of ( C , D )-bimodules with the model structure discussedin [BM12, 2.4-2.8], in which the weak equivalences are the pointwise equivalences.Bimodules over spectral categories form a bicategory with shadow, and this structureis the foundation of all our work in this paper. It echoes the structure of the bicategoryof rings and bimodules from Example 2.5. There are several previous constructions inthe literature, for instance [Shu06, 22.11], [LM19, 4.13], [CP19, 2.17], [Mal19, 7.4.2]. Definition 4.9.
Let
Bimod ( SpCat ) be the bicategory with the pointwise cofibrant spectral categories C ,
1- and 2-cells the objects and morphisms in the homotopy categories Ho ¡ Mod ( C , D ) ¢ ,and horizontal composition of 1-cells X , a ( C , D )-bimodule, and Y , a ( D , E )-bimodule, given by the bar construction X ⊙ Y : = B ( X , D , Y ).The bimodules D = D id id are the units for the horizontal composition, with unit iso-morphisms given by (4.7). The associativity of the horizontal composition follows froma comparison of bisimplicial spectra. We equip the bicategory Bimod ( SpCat ) with ashadow using topological Hochschild homology: 〈〈 X 〉〉 : = THH( C ; X ).The horizontal composition of 1-cells is compatible with the bimodule structures givenby functors: -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 17 Lemma 4.10.
For any twisting C F G there is a canonical isomorphism of 1-cells C F G ≃ C F ⊙ C G . For composable spectral functors A → B → C , there are canonical isomorphisms of 1-cells B A ⊙ C B ≃ C A C B ⊙ B A ≃ C A .Our examples of 1-cells also give simple ways to construct dual pairs: Proposition 4.11 ([Pon10, Appendix], [PS12, Lem. 7.6]) . Let F : A → C be a spectralfunctor. Then there is a dual pair ( C F , C F ) whose coevaluation and evaluation maps areinduced by the composites η : A ( a , b ) F −−−→ C ( F a , F b ) ∼ −→ B ( C , C , C )( F a , F b ) = ( C F ⊙ C F )( a , b ) and ǫ : ( C F ⊙ C F )( c , d ) = B ( C F , A , C F )( c , d ) F −−−→ B ( C , C , C )( c , d ) ∼ −−→ C ( c , d ).4.2. Bicategorical traces and
THH . In this subsection we show how familiar maps onTHH can be described as traces of endomorphisms of bimodules. The primary results arewell-known, but our explicit use of the shadow structure on THH simplifies and clarifiesprevious proofs, e.g. in the work of Blumberg and Mandell [BM12].
Proposition 4.12 ([CP19, 5.8]) . Let F : A → C be a spectral functor.i . Given a ( C , C ) -bimodule X , write X F F : = C F ⊙ X ⊙ C F . Then the trace of the map X F F ⊙ C F = C F ⊙ X ⊙ C F ⊙ C F id ⊙ ǫ −−−→ C F ⊙ X , taken with respect to the dual pair ( C F , C F ) , is the map THH( F ) : THH( A ; X F F ) −→ THH( C ; X ) induced by F on the cyclic bar construction.ii . The Euler characteristic χ ( C F ) of the left dualizable 1-cell C F (resp. the rightdualizable 1-cell C F ) is the induced map THH( F ) : THH( A ) −→ THH( C ). Definition 4.13.
A spectral functor F : A → C is a Morita equivalence if the dual pair( C F , C F ) is a Morita equivalence, in the sense of Definition 2.15. Lemma 4.14 ([BM12, 5.12]) . If F is a Dwyer–Kan embedding and surjective up to thickclosure, then F is a Morita equivalence.
The condition of being surjective up to thick closure means that the representablefunctors C ( c , − ) can be obtained from the representable functors C ( F a , − ) for a ∈ ob A bycofiber sequences and retracts, and similarly on the other side C ( − , c ).Proposition 4.12 implies that when F : A → C is a Morita equivalence, the inducedmap THH( F ) is an equivalence. The next result follows immediately: Theorem 4.15 (See [CP19, 5.9] and [BM12, 5.12]) . Let F : A → C be a map of pointwisecofibrant spectral categories and X be a ( C , C ) -bimodule. If F is a Dwyer–Kan embeddingand surjective up to thick closure, then the induced maps of spectra THH( F ) : THH( A ) −→ THH( C ), THH( F ) : THH( A ; X F F ) −→ THH( C ; X ) are equivalences. We can use the theorem to show that THH( A ) is equivalent to THH( P A ). Example 4.16. If A is a ring spectrum, then the inclusion of spectral categories A −→ P A is a Dwyer–Kan embedding, and surjective up to thick closure essentially by thedefinition of P A . Thus THH( A ) ∼ −→ THH( P A ).Furthermore, if M is an ( A , A )-bimodule, then along the map A → P A we have an equiv-alence of bimodules M → M A M from Example 4.5, and hence an equivalenceTHH( A ; M ) ∼ −→ THH( P A ; M A M ). Example 4.17.
In Lemma 3.15 we saw that the iterated degeneracy maps w C → w k C for a spectral Waldhausen category C are Dwyer–Kan equivalences, and thus induceequivalences on THH. Example 4.18.
Let C and D be spectral categories with chosen zero object, let C × D betheir product in spectral categories, and let C ∨ D ⊂ C × D be the full subcategory spannedby the pairs in which at least one coordinate is the zero object. Then the inclusion C ∨ D −→ C × D is both a Dwyer–Kan embedding and surjective up to thick closure. Thereforewe have an equivalenceTHH( C ) ∨ THH( D ) ∼= −→ THH( C ∨ D ) ∼ −→ THH( C × D ).This gives a short proof of the nontrivial fact that THH preserves finite products.We can further strengthen Proposition 4.12 if X is of the form D F G . Proposition 4.19.
Given a morphism of twistings ( I , J ) : C / D F G → C ′ / D ′ F ′ G ′ there areinduced maps D F G ⊙ C ′ I → C ′ I ⊙ D ′ F ′ G ′ and C ′ I ⊙ D F G → D ′ F ′ G ′ ⊙ C ′ I whose traces are both the induced map THH( I ; J ) : THH( C ; D F G ) −→ THH( C ′ ; D ′ F ′ G ′ ) Proof.
Write b F : = JF and b G : = JG , and observe that b F = F ′ I and b G = G ′ I by the commu-tativity of the diagram defining a morphism of twistings (Definition 2.17). We prove theproposition for the first map; the second is proved analogously.The desired map is the composite D F ⊙ D G ⊙ C ′ C id ⊙ η ⊙ id −−−−−→ D F ⊙ D ′ D ⊙ D ′ D ⊙ D G ⊙ C ′ C ∼ −→ D ′ b F b G ∼ −→ C ′ C ⊙ D ′ F ′ ⊙ D ′ G ′ ⊙ C ′ C ⊙ C ′ C id ⊙ ǫ −−−→ C ′ C ⊙ D ′ F ′ ⊙ D ′ G ′ ,where the two isomorphisms of 1-cells are obtained using Lemma 4.10. This map isnow in a form where its trace agrees with the right-hand side of the equation in [PS13,Proposition 7.1], with M = C ′ C . Applying the proposition and simplifying implies thatthe trace of this map is 〈〈 D F G 〉〉 〈〈 id ⊙ η ⊙ id 〉〉−−−−−−−−→ 〈〈 D ′ b F b G 〉〉 tr(id ⊙ ǫ ) −−−−−−→ 〈〈 D ′ F ′ G ′ 〉〉 .Applying the definition of the shadow 〈〈−〉〉 on the bicategory of spectral categories andusing Proposition 4.12 with X = D ′ F ′ G ′ , the composite isTHH( C ; D F G ) THH(id C ; J ) −−−−−−−−→ THH( C ; D ′ b F b G ) THH( I ) −−−−−→ THH( C ′ ; D ′ F ′ G ′ ),where the first map applies J to mapping spectra in D and the second is the map inducedby I . (cid:3) Putting Theorem 4.15 and Proposition 4.19 together gives the following: These are examples of
Beck–Chevalley maps . -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 19 Corollary 4.20.
Let ( I , J ) : C / D F G → C ′ / D ′ F ′ G ′ be a morphism of twistings where I andJ are Dwyer–Kan embeddings and I is surjective up to thick closure. Then the inducedmap THH( I ; J ) : THH( C ; D F G ) −→ THH( C ′ ; D ′ F ′ G ′ ) is an equivalence.
5. T
HE ADDITIVITY THEOREM FOR
THH,
REVISITED
Additivity without coefficients.
In this section we prove:
Theorem 5.1.
There is an equivalence of spectra ( ι j ) kj = : k _ j = THH( C ) ∼ −→ THH( S k C ).This is similar in spirit to existing additivity results, such as [DM96, 1.6.20] and[DGM13, IV.2.5.8] which use a category of upper-triangular matrices, [BM12, Thm. 10.8]which proves a version for DG-categories, and [BM11, 3.1.1] which proves additivity forWTHH( C ) : = THH( S • C ), in other words after one copy of S • has been applied.Our approach to Theorem 5.1 is fundamentally an adaptation of a technique from[BM12, §7], made more conceptual by the machinery of shadows. Definition 5.2.
Let C be a spectral Waldhausen category. By [CLM + , Theorem 4.1] thereis a canonical equivalence S C ≃ C . Let s k − : S k − C → S k C and d k : S k C → S k − C be thelast degeneracy and face functors, respectively. Let π k : S k C → S C ∼ → C be the inducedby d k − and let ι k : C ∼ → S C → S k C be induced by s k − . More generally, for 1 ≤ j ≤ k write ι j : C → S k C for the functor induced by s j − s k − j ; these are the functors inducingthe equivalence in Theorem 5.1.The next proposition is the main ingredient for the proof of the additivity theorem. Asthe proof is technical, we postpone it until §5.3. Proposition 5.3.
The coevaluation map of the dual pair ( ( S k C ) s k − , ( S k C ) s k − ) and theevaluation map of the dual pair ( C π k , C π k ) are pointwise equivalences of bimodules. Theother evaluation map and coevaluation map induce a homotopy cofiber sequence of ( S k C , S k C ) -bimodules ( S k C ) s k − ⊙ ( S k C ) s k − ǫ −→ S k C η −−→ C π k ⊙ C π k .Theorem 5.1 follows by induction from the following proposition: Proposition 5.4.
The spectral functors s k − and ι k induce an equivalence THH( S k − C ) ∨ THH( C ) ∼ −→ THH( S k C ).Figure 5.5 is the version of Figure 2.9a for Theorem 5.1. Proof.
By Proposition 4.12, the induced map THH( s k − ) is the Euler characteristic of( S k C ) s k − , computed using the dual pair ( ( S k C ) s k − , ( S k C ) s k − ). By Definition 2.14, theEuler characteristic is the following composite: χ (( S k C ) s k − ) : 〈〈 S k − C 〉〉−→ 〈〈 ( S k C ) s k − ⊙ ( S k C ) s k − 〉〉≃ 〈〈 ( S k C ) s k − ⊙ ( S k C ) s k − 〉〉 ǫ −→ 〈〈 S k C 〉〉 .The first map is an equivalence by the first statement in Proposition 5.3. Rewriting interms of THH, it follows that the induced map THH( s k − ) is the composite(5.6) THH( S k − C ) ∼ −→ THH( S k C ; ( S k C ) s k − ⊙ ( S k C ) s k − ) ǫ −→ THH( S k C ).Similarly, the induced map THH( π k ) is the composite(5.7) χ ( C π k ) : THH( S k C ) η −−→ THH( S k C ; C π k ⊙ C π k ) ∼ −→ THH( C ),where the equivalence is induced by the evaluation map of the dual pair ( C π k , C π k ). ( A ) THH( S k C ) ( B ) W ki = THH( C ) F IGURE S k C ; − ) to the cofiber sequence from Proposition 5.3. Since THH preserves cofibersequences, this produces a cofiber sequenceTHH( S k − C ) s k − −→ THH( S k C ) π k −→ THH( C ).The second map has a section, induced by ι k , so the cofiber sequence splits. (cid:3) Additivity with coefficients.
Next we generalize Theorem 5.1 to allow for twistedcoefficients. For ease of future reference we state the theorem in its multisimplicial form.
Remark . From the properties of the S • -construction, any twisting C / D L R of spectralWaldhausen categories induces a twisting ( S • C / S • D ) S • L S • R . For ease of reading, in suchcases we drop the S • -notation from the subscripts and simply write ( S • C / S • D ) L R . Theorem 5.9.
Given a twisting C / D L R of spectral Waldhausen categories there is anequivalence of spectra _ ≤ i j ≤ k j ≤ j ≤ n THH( C ; D L R ) ∼ −→ THH( w k S ( n ) k ,..., k n C ; ( w k S ( n ) k ,..., k n D ) L R ).The theorem follows from the w • -invariance of THH (see Example 4.17), and an in-ductive argument based on the following generalization of Proposition 5.4: Proposition 5.10.
Let C / D L R be a twisting of spectral Waldhausen categories. The spec-tral functors s k − and ι k induce an equivalence THH( S k − C ; ( S k − D ) L R ) ∨ THH( C ; D L R ) ∼ −→ THH( S k C ; ( S k D ) L R ).The proof of this proposition is largely analogous to the proof of Proposition 5.4; themain difficulty that the twisting adds is that the construction of the equivalences in (5.6)and (5.7) requires an extra step.The functors s k − , d k , π k , and ι j are functors of spectral Waldhausen categories. Bydefinition, d k s k − = id and π k ι k = id. These identities define the unit and counit, respec-tively, of adjunctions S k − C s k − / / S k C d k o o S k C π k / / C ι k o o on the associated base categories. These adjunctions do not extend to spectrally enrichedadjunctions between our models for the spectral categories S k C , because the Moore endis not 2-functorial ([CLM + , Section 4.5]). They do, however, still satisfy a conditionanalogous to an adjunction: -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 21 Proposition 5.11.
The spectral functors d k and π k induce equivalences of spectraS k C ( s k − a , b ) ∼ −→ S k − C ( d k s k − a , d k b ) = S k − C ( a , d k b ) S k C ( a , ι k b ) ∼ −→ C ( π k a , π k ι k b ) = C ( π k a , b ).We postpone the proof of the proposition to §5.3, and now prove Proposition 5.10. Proof of Proposition 5.10.
We fill in the details that differ from the proof of Proposi-tion 5.4. The spectral functor s k − defines a morphism of twistings( s k − , s k − ) : ( S k − D ) L R −→ ( S k D ) L R .By Proposition 4.19 there is an associated map f : ( S k − D ) L R ⊙ ( S k C ) s k − −→ ( S k C ) s k − ⊙ ( S k D ) L R whose trace is the induced mapTHH( s k − ) : THH( S k − C ; ( S k − D ) L R ) −→ THH( S k C ; ( S k D ) L R ).By definition, the trace of f is the composite 〈〈 ( S k − D ) L R 〉〉 η −−→ 〈〈 ( S k − D ) L R ⊙ ( S k C ) s k − ⊙ ( S k C ) s k − 〉〉 f −−→ 〈〈 ( S k C ) s k − ⊙ ( S k D ) L R ⊙ ( S k C ) s k − 〉〉∼= 〈〈 ( S k C ) s k − ⊙ ( S k C ) s k − ⊙ ( S k D ) L R 〉〉 ǫ −→ 〈〈 ( S k D ) L R 〉〉 .As in the proof of Proposition 5.4, we will prove that, prior to the application of theevaluation map, the composite is an equivalence. It suffices to prove that f is a pointwiseequivalence of ( S k − C , S k C )-bimodules, which follows from the commutative diagram( S k − D ) L ⊙ ( S k − D ) R ⊙ ( S k C ) s k − ( S k C ) s k − ⊙ ( S k D ) L ⊙ ( S k D ) R ( S k − D ) L ⊙ ( S k D ) s k − ⊙ ( S k D ) R ( S k − D ) L ⊙ ( S k − D ) R ⊙ ( S k − C ) d k ( S k − D ) L ⊙ ( S k D ) d k ⊙ ( S k D ) R , f ≃ id ⊙ id ⊙ d k ≃≃ id ⊙ d k ⊙ id ≃ where the unlabeled equivalences are instances of Lemma 4.10 and the equivalencesinduced by d k are from Proposition 5.11.Similarly, the map π k induces a morphism of twistings( π k , π k ) : ( S k D ) L R → D L R .The same logic as above reduces the proof to showing that the map g : ( S k D ) L R ⊙ C π k −→ C π k ⊙ D L R from Proposition 4.19 is a pointwise equivalence of bimodules, which follows in the samemanner from the commutative diagram( S k D ) L ⊙ ( S k D ) R ⊙ ( S k C ) ι k ( S k D ) L ⊙ ( S k D ) ι k ⊙ D R ( S k D ) L ⊙ D π k ⊙ D R ( S k D ) L ⊙ ( S k D ) R ⊙ C π k C π k ⊙ D L ⊙ D R . ≃ id ⊙ id ⊙ π k ≃ id ⊙ π k ⊙ id ≃≃ g The rest of the proof proceeds as in the untwisted case. (cid:3)
Remark . These results can be generalized to the case when D is a pointed spectralcategory. In this case, w k D is defined to include all maps and w D → w k D is still aDwyer–Kan embedding. The definition of S k D from §3.3 is modified to be the subcate-gory of Fun([ k ] × [ k ], D ) on diagrams sending each ( i ≥ j ) to the zero object ∗ ∈ D .This doesn’t affect any of the proofs because we only ever consider diagrams in theimage of the functors L and R , and so it is enough to control the behavior of cofibrationsand pushouts in S k C .5.3. The technical proofs.
In this subsection we prove Propositions 5.3 and 5.11.
Proof of Proposition 5.11.
The proof requires explicit properties of the construction of themapping spectra in S k C from [CLM + , §4]. The key fact is that the mapping spectrum S k C ( a , b ) is equivalent, via the canonical restriction maps, to the homotopy limit of thezig-zag diagram of spectra built out of the composition maps between C ( a ( i , j ), b ( i , j ))for 0 ≤ i , j ≤ k [CLM + , Theorem 4.1 ii ]. Under these equivalences, the last face functor d k : S k C → S k − C agrees with the map to the homotopy limit of the subdiagram where0 ≤ i , j ≤ k −
1. Replacing the domain a with a diagram s k − a in the image of the lastdegeneracy functor, the canonical map s k − a ( k − j ) → s k − a ( k , j ) is the identity map,and thus the induced maps of spectra C ( s k − a ( k , j ), b ( k , j )) −→ C ( s k − a ( k − j ), b ( k , j ))are identity maps for any j . Similarly, restricting to the bottom row gives C ( s k − a ( i , k ), b ( i , k )) = ∗ for any i . It follows that the map of homotopy limits from the diagram with 0 ≤ i , j ≤ k to the diagram with 0 ≤ i , j ≤ k − d k induces an equivalence of mapping spectra S k C ( s k − a , b ) ≃ −→ S k − C ( d k s k − a , d k b ),as claimed. A similar argument shows that the functor π k induces an equivalence ofmapping spectra S k C ( a , ι k b ) ∼ −→ C ( π k a , π k ι k b ). (cid:3) For ease of notation, we require an extra definition.
Definition 5.13.
A pointwise map of ( C , D )-bimodules, denoted X / / Y , is a map ofspectra X ( c , d ) → Y ( c , d ) for each c ∈ ob C and d ∈ ob D . These are not required to satisfyany coherence with the C and D actions. When a pointwise map of ( C , D )-bimodules iscompatible with the C -action, we denote it by X (cid:31) (cid:127) / / Y .We write ǫ : s k − d k −→ id for the counit of the adjunction ( s k − ⊣ d k ) of base categor-ies. Composing with the morphism ǫ : s k − d k b −→ b in S k C defines a pointwise map ofbimodules ( S k C ) s k − d k (cid:31) (cid:127) / / S k C ,compatible with the left action. Similarly, composing with the unit η : b −→ ι k π k for theadjunction ( π k ⊣ ι k ) of base categories defines a pointwise map of bimodules( S k C ) (cid:31) (cid:127) / / ( S k C ) ι k π k that is also compatible with the left action. -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 23 Lemma 5.14.
The pointwise equivalences from Proposition 5.11 fit into commutativediagrams of pointwise morphisms of bimodules ( S k C ) s k − ⊙ ( S k C ) s k − ≃ (cid:15) (cid:15) ǫ / / S k C ( S k C ) s k − ⊙ ( S k − C ) d k ≃ / / ( S k C ) s k − d k ?(cid:31) ǫ O O and S k C η / / (cid:127) _ η (cid:15) (cid:15) C π k ⊙ C π k ( S k C ) ι k π k ≃ / / ( S k C ) ι k ⊙ C π k ≃ O O relating the evaluation (resp. coevaluation) map for the dual pairs with the counit (resp.unit) of the adjunction on base categories.Proof. We prove the lemma for the first diagram; the second follows analogously. Recallthat the spectral category S k C is defined as a full subcategory of the functor categoryFun([ k ] , C ). Define a twisting of spectral categories S k C / ‚ S k C L R where ‚ S k C ⊆ Fun([1] × [ k ] , C )is the full subcategory of diagrams that at each i ∈ [1] satisfy the conditions for S k C .The spectral functor L : S k C → ‚ S k C arises from the collapse [1] → ∗ , and the spectralfunctor R : S k C → ‚ S k C arises from the map of posets [1] × [ k ] → [ k ] that on 1 ∈ [1] isthe identity of [ k ] and on 0 ∈ [1] applies i max( i , k −
1) to each coordinate of [ k ] (thisis the map of totally ordered sets inducing s k − d k ). Let r , r : ‚ S k C ⇒ S k C denote thespectral functors that restrict to 0 ∈ [1] and 1 ∈ [1], respectively. We form the followingdiagram of ( S k C , S k C )-bimodules S k C s k − ⊙ S k C s k − (1,1, d k ) (cid:15) (cid:15) = * * ‚ S k C s k − ⊙ S k C s k − ∼ ( r ,1,1) o o (1, s k − ,1) (cid:15) (cid:15) ∼ ( r ,1,1) / / S k C s k − ⊙ S k C s k − (1, s k − ,1) (cid:15) (cid:15) S k C s k − ⊙ S k − C d k ∼ (cid:15) (cid:15) ‚ S k C ⊙ S k C ∼ ( r , d k , d k ) o o ∼ (cid:15) (cid:15) ∼ ( r ,1,1) / / S k C ⊙ S k C ∼ (cid:15) (cid:15) ( S k C ) s k − d k (cid:24) x ǫ ‚ S k C ∼ r o o ∼ r / / S k C .The four rectangular regions commute and all the solid arrows are maps of ( S k C , S k C )-bimodules, as is verified by checking that various maps of twistings, arising from mapsof posets, agree with each other. The top region also commutes easily. The region atthe very bottom commutes in the category of pointwise maps of bimodules, again usingthe description of Fun([1], C ) as a homotopy limit of a zig-zag – see [CLM + , §4] for moredetails. The outside maps are the desired pointwise maps of bimodules, finishing theproof. (cid:3) We are now ready to prove Proposition 5.3.
Proof.
The first claim in the proposition follows from the commutativity of the diagrams S k − C ( S k C ) s k − ⊙ ( S k C ) s k − ( S k − ) d k s k − ( S k − C ) d k ⊙ ( S k C ) s k − η d k ⊙ id ≃≃ and C π k ⊙ C π k CC π k ⊙ ( S k C ) ι k C π k ι k ǫ ≃ id ⊙ π k ≃ which are formally analogous to those in Lemma 5.14 (but easier to check because η = idfor ( s k − ⊣ d k ) and ǫ = id for ( π k ⊣ ι k )).To check the cofiber sequence statement note that the two diagrams in Lemma 5.14give, for each x , y ∈ ob S k C , a natural weak equivalence betwen the sequence of interestand the sequence S k C ( x , s k − d k y ) → S k C ( x , y ) → S k C ( x , ι k π k y ).Thus to show that the given sequence of bimodules is a homotopy cofiber sequence itsuffices to show that this is a homotopy cofiber sequence of spectra for each x , y .The counit and unit of the adjunctions ( s k − ⊣ d k ) and ( π k ⊣ ι k ) of base categories fitinto a pushout square of functors S k C → S k C ( s k − d k ) (cid:15) (cid:15) ǫ / / id η (cid:15) (cid:15) ∗ / / ( ι k π k )whose horizontal arrows are cofibrations. Since S k C is a spectral Waldhausen category,there is an induced homotopy pushout square of spectra S k C ( x , s k − d k y ) (cid:15) (cid:15) / / S k C ( x , y ) (cid:15) (cid:15) S k C ( x , ∗ ) / / S k C ( x , ι k π k y ).Since the lower-left corner is contractible, the other three terms form a homotopy cofibersequence, as desired. (cid:3)
6. T HE D ENNIS TRACE
In this section we construct the Dennis trace map K (End( C )) → THH( C ) out of endo-morphism K -theory for a spectrally enriched Waldhausen category C , as well as a twistedDennis trace which allows bimodule coefficients. This material serves as the scaffoldingfor the construction of the trace map to TR in §7–8. We conclude the section with a con-crete description of the effect of the Dennis trace on π in terms of bicategorical traces(Proposition 6.24).6.1. Endomorphism categories.
We begin by defining endomorphism categories.
Definition 6.1.
For any Waldhausen category C , let End( C ) be the Waldhausen cat-egory of functors Fun( N , C ), where N is considered as a category with one object andmorphism set N . More concretely, the objects of End( C ) are endomorphisms f : a −→ a in C , and the morphisms are commuting squares of the form a f / / i (cid:15) (cid:15) a i (cid:15) (cid:15) b g / / b .We define the morphism to be a cofibration or weak equivalence if i is a cofibration orweak equivalence, respectively. We also define exact functors C End( C ) ι ι -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 25 where End( C ) −→ C forgets the endomorphism f . The inclusions ι , ι : C −→ End( C )equip each object a with either the zero endomorphism or the identity endomorphism. Example 6.2. If A is a ring spectrum and C = P = P A is the spectral Waldhausen cate-gory of perfect A -modules from Example 3.10, then K ( C ) is the usual algebraic K -theoryspectrum K ( A ) of A , and the K -theory of End( C ) is the K -theory of endomorphisms K (End( A )).It is also possible to extend the definition of endomorphism K -theory to twistings.First we recall our main example of a twisting. Example 6.3.
Let A be a ring spectrum. Recall from Definition 3.5 and Examples 3.6and 4.5 the spectral categories P of perfect left A -modules and M of all left A -modules.Let L , R : P ⇒ M denote, respectively, the inclusion and the functor M ∧ A − for a cofibrant ( A , A )-bimodule M . This defines a twisting that we denote by P / M M . Definition 6.4.
Given a twisting C / D L R of spectral Waldhausen categories, the twistedendomorphism category
End ¡ C / D L R ¢ is the category where • the objects are pairs ( a , f ) of a ∈ ob C and a morphism f : L ( a ) → R ( a ) in D , and • a morphism ( a , f ) → ( b , g ) is a morphism i : a → b in C such that the diagram L ( a ) R ( a ) L ( b ) R ( b ) fL ( i ) R ( i ) g commutes.Note that this definition only uses the base categories C and D , and not the spectralenrichment. Example 6.5. i . When D = C and L = R = id C we get the usual endomorphism category. ii . The twisted endomorphism category for P / M M = P A / M A M has as objects A -module maps P → M ∧ A P with P a perfect A -module. Following [LM12], the K -theory of its base Waldhausen category is denoted by K ( A ; M ) : = K End( P A / M A M ).6.2.
Bispectra.
Before defining the Dennis trace, we introduce some formal structurethat arises naturally when analyzing THH.
Definition 6.6. A bispectrum is a symmetric spectrum object in orthogonal spectra[MMSS01, Hov01, CLM + ].In order to construct bispectra, we need a technical tool which formalizes the way thatthe iterated S • -constructions | S ( n ) • , ··· , • C | assemble into a symmetric spectrum. Let I bethe skeleton of the category of finite sets and injections spanned by the objects n = {
1, . . ., n } for n ≥
0. Let ∆ op × n be the n -fold product of the opposite of the category ∆ of nonemptytotally ordered finite sets [ k ] = { < · · · < k } .For each morphism f : m −→ n in I , there is an induced functor f ∗ : ∆ op × m −→ ∆ op × n taking ([ k ], . . ., [ k m ]) to the n -tuple whose value at f ( i ) is [ k i ] and whose value outsidethe image of f is always [1]. In particular, when m = n there is an action of the symmetric group Σ n on ∆ op × n . This rule defines a strict diagram of categories indexed by I , andwe write I R ∆ op ×− for its Grothendieck construction. Thus, the objects of the category I R ∆ op ×− are tuples ( m ; k , . . ., k m ),where m , k i ≥
0, and a morphism( m ; k , . . ., k m ) → ( n ; l , . . ., l n )consists of an injection f : m −→ n and a morphism ( φ i ) : f ∗ ([ k ], . . ., [ k m ]) → ([ l ], . . ., [ l n ])in ∆ op × n . Definition 6.7.
Given a pointed category M , a Σ ∆ -diagram in M is a functor X ( • ; • ,..., • ) : I R ∆ op ×− −→ M with the following two properties: • X ( m ; k ,..., k m ) ∼= ∗ any time k i = i , and • the morphisms ( m ; k , . . ., k m ) −→ ( n ; f ∗ ( k , . . ., k m )) with every φ i = id induce iso-morphisms X ( m ; k ,..., k m ) ∼= X ( n ; f ∗ ( k ,..., k m )) .The symmetric group action on ∆ op × n defines an action of Σ n on the geometric realiza-tion of the multisimplicial object | X ( n , • ,..., • ) | , and this construction extends to a functorfrom Σ ∆ -diagrams in a pointed simplicial model category M to symmetric spectrum ob-jects in M [CLM + , Lemma 6.3]. For further discussion of Σ ∆ -diagrams, see [CLM + , §6].6.3. Definition of the Dennis trace.
Let C be a spectral Waldhausen category and let X be a ( C , C )-bimodule. The key observation for the construction of the Dennis trace isthat the inclusion of 0-simplices in the cyclic bar construction defines a canonical map(6.8) _ c ∈ ob C X ( c , c ) → THH( C ; X ).When X = C , each object f : c → c of End( C ) defines a map of spectra S −→ C ( c , c ),and so composing with (6.8) gives a map(6.9) Σ ∞ ob End( C ) = _ f : c → c , c S −→ _ c ∈ ob C C ( c , c ) −→ THH( C )where f runs over the objects of End( C ). See Figure 6.11a for a picture of this map.Applying (6.9) to the spectral Waldhausen category w k S ( n ) k ,..., k n C for each value of n and k , . . ., k n defines a map of orthogonal spectra Σ ∞ ob End( w k S ( n ) k ,..., k n C ) −→ THH( w k S ( n ) k ,..., k n C ).Appending the splitting from Theorem 5.9 gives a zig-zag of orthogonal spectra(6.10) Σ ∞ ob End( w k S ( n ) k ,..., k n C ) −→ THH( w k S ( n ) k ,..., k n C ) ≃ ←− _ i ,..., i n ≤ i j ≤ k j THH( C ).The number of summands on the right is the same as the number of nonzero points inthe set S k ∧· · ·∧ S k n , where S • is the simplicial circle ∆ [1]/ ∂ ∆ [1]. Therefore these wedgesums form an ( n + k direction.The construction of the zig-zag (6.10) works identically for a bimodule arising froma twisting C / D L R of spectral Waldhausen categories. In this case, the map (6.8) with X = ( w • S ( n ) • D ) L R defines a zig-zag of multisimplicial orthogonal spectra(6.12) Σ ∞ ob End( ( w • S ( n ) • C / w • S ( n ) • D ) L R ) −→ THH( w • S ( n ) • C ; ( w • S ( n ) • D ) L R ) ≃ ←− ( S • ) ∧ n ∧ THH( C ; D L R ). -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 27 c c c f CC ( A ) C / D L R = C c R ( c ) L ( c ) f L D R C ( B ) General case F IGURE
Lemma 6.13.
The maps in the zig-zag (6.12) of multisimplicial orthogonal spectra com-mute with the Σ n -actions and the identifications that remove a simplicial direction whenits index is equal to 1. In other words the given maps produce a zig-zag of Σ ∆ -diagramsof simplicial orthogonal spectra. Another way of saying this is that ( S • ) ∧ n ∧ THH( C ) is the free Σ ∆ -diagram on thespectrum THH( C ) at level (0; ), and W ι i ,..., i k agrees with the map that arises from thefree-forgetful adjunction.Taking the geometric realization of these multisimplicial orthogonal spectra gives azig-zag of bispectra. At level n in the symmetric spectrum direction, the zig-zag of bis-pectra is(6.14) | Σ ∞ ob End( ( w • S ( n ) • C / w • S ( n ) • D ) L R ) | −→ | THH( w • S ( n ) • C ; ( w • S ( n ) • D ) L R ) | ≃ ←− Σ n THH( C ; D L R ).There is a canonical identification of setsob End( ( w • S ( n ) • C / w • S ( n ) • D ) L R ) = ob w • S ( n ) • End( C / D L R ).This identifies the bispectrum on the left of (6.14) with the orthogonal suspension spec-trum of the symmetric spectrum K (End( C / D L R )). On the other hand, the spectrum onthe right of (6.14) is the symmetric suspension spectrum of the orthogonal spectrumTHH( C ; D L R ).Applying the (left-derived) prolongation functor from [CLM + , Proposition A.7] to thesebispectra, we get a zig-zag of orthogonal spectra(6.15) P K (End( C / D L R )) −→ P | THH( w • S ∗• C ; ( w • S ∗• D ) L R ) | ≃ ←− THH( C ; D L R ).The first term in (6.15) is the prolongation of K (End( C / D L R )) from symmetric to orthog-onal spectra.
Definition 6.16.
The
Dennis trace map associated to a twisting C / D L R is obtained bychoosing an inverse to the wrong-way map in (6.15), defining a map(6.17) trc : P K (End( C / D L R )) −→ THH( C ; D L R )in the homotopy category of orthogonal spectra, or, equivalently, in the homotopy cate-gory of symmetric spectra(6.18) trc: K (End( C / D L R )) −→ U THH( C ; D L R )to the underlying symmetric spectrum of THH.
Remark . It is unnecessary to check that the multisimplicial objects above are Reedycofibrant, because the realization can be automatically left-derived without losing thesymmetric spectrum structure ([CLM + , Theorem 6.4]). See [CLM + , Remark 7.14] for additional discussion. The upshot is that this construction of the Dennis trace is insen-sitive to the choice of model for THH. Observation 6.20.
Consider the case when C = D and L = R = id . The Dennis trace mapon K (End( C )) extends the Dennis trace on K ( C ) , as constructed in e.g. [BM11, DGM13] ,in the sense that the following diagram commutes:K ( C ) trc (cid:15) (cid:15) ι / / K (End( C )) trc w w ♦♦♦♦♦♦♦♦♦♦♦ THH( C ).To see this, observe that other authors construct the Dennis trace map in the sameway that we did for K (End( C )), except using the map (6.8) with X ( c , c ) = S instead of(6.9). The two maps agree along the inclusion of the identity morphisms, and tracingthrough the construction above this becomes the map of K -theory spectra ι : K ( C ) −→ K (End( C )) induced by the exact functor ι from beneath Definition 6.1. Remark . The same argument shows that the diagram K ( C ) (cid:15) (cid:15) ι / / K (End( C )) trc w w ♣♣♣♣♣♣♣♣♣♣♣ THH( C )commutes. As a result, the Dennis trace factors through the mapping cone of ι , oftencalled cyclic K -theory (or the reduced K -theory of endomorphisms): K cyc ( C ) = e K (End( C )) = K (End( C )) / ι K ( C ).If we return to the case of general D , the inclusion of identity endomorphisms ι becomes meaningless, but the inclusion of zero endomorphisms ι is still defined, andthere is a commutative diagram K ( C ) (cid:15) (cid:15) ι / / K End ¡ C / D L R ¢ trc v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ THH ¡ C ; D L R ¢ .Thus the Dennis trace descends to a map out of cyclic K -theory, e K End ¡ C / D L R ¢ = K End ¡ C / D L R ¢ / ι K ( C ) −→ THH ¡ C ; D L R ¢ .In particular, for a ring spectrum A and bimodule M the Dennis trace defines a map K cyc ( A ; M ) = e K ( A ; M ) trc −−→ THH( P A ; M A M ) ∼ ←− THH( A ; M ). Remark . We sketch an argument that this agrees with the trace defined in [LM12,9.2], see also [DGM13, V]. The idea is to use a version of [DMP +
19] with bimodule co-efficients to turn our smash products into Bökstedt smash products. Then the relevantbispectrum is an Ω -spectrum in the THH direction, hence the prolongation is equivalentto the functor that restricts to the symmetric spectrum direction. After these manip-ulations, the inclusion of endomorphisms map (6.9) is the same as the one in [LM12,9.1]. -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 29 Example 6.23.
Taking C = P A for a ring spectrum A , we get maps K ( A ) e K End( A ) THH( A ) ∼ (cid:15) (cid:15) K ( P A ) ι / / e K End( P A ) trc / / THH( P A )whose composite agrees with the Dennis trace map K ( A ) → THH( A ) studied previously,e.g. [DM96, DGM13, Mad94]. Proposition 6.24.
Let f : L ( a ) → R ( a ) be an object of the twisted endomorphism category End ¡ C / D L R ¢ . The image of [ f ] ∈ K End ¡ C / D L R ¢ under the Dennis trace is the homotopyclass of the composite S [ f ] −−→ D ( L ( a ), R ( a )) −→ _ c ∈ C D ( L ( c ), R ( c )) -skeleton −−−−−−−→ THH( C ; D L R ) which includes f as a -simplex in the cyclic bar construction.Proof. Consider the following commutative diagram, where the top row maps to thebottom row by mapping symmetric spectrum level 0 and simplicial level 0 into the zig-zag of bispectra: Σ ∞ ob End( w C / D L R ) (cid:15) (cid:15) / / THH( w C ; w D L R ) (cid:15) (cid:15) Σ THH( C ; D L R ) ∼= o o ∼= (cid:15) (cid:15) P K (End( C / D L R )) / / P | THH( w • S ∗• ,..., • C ; ( w • S ∗• ,..., • D ) L R ) | THH( C ; D L R ) ∼= o o The vertical map on the left is surjective on π by the standard presentation for K of aWaldhausen category. By the construction of the Dennis trace, the first horizontal mapin the top row is the inclusion of endomorphisms map, so the conclusion follows. (cid:3) Example 6.25. If A is a ring spectrum, each perfect left A -module P defines a class[ P ] ∈ K ( A ). By Proposition 6.24, its image in π THH( P A ) is the inclusion of the 0-simplex corresponding to the identity map of P into the cyclic bar construction. By[CP19, §7], the image of this class under the Morita equivalence π THH( P A ) ∼= π THH( A )is the Euler characteristic of P . More generally, each endomorphism f : P −→ P definesa class [ f ] ∈ K End( A ) whose image in π THH( A ) is the trace of f , by Proposition 6.24and [CP19, 7.11]. Example 6.26.
Every twisted endomorphism f : P → M ∧ A P of a perfect left A -module P defines a class [ f ] ∈ K ( A ; M ) whose image in π THH( P A ; M A M ) is the inclusion of the0-simplex corresponding to f in the cyclic bar construction. By [CP19, 7.4], applied as inthe proof of [CP19, 7.11], its image under the Morita equivalence π THH( P A ; M A M ) ∼= π THH( A ; M )is the bicategorical trace tr( f ) : S = 〈〈 S 〉〉−→ 〈〈 M 〉〉= THH( A ; M ).
7. T
HE EQUIVARIANT D ENNIS TRACE
In order to build a trace to topological restriction homology (TR), we construct C r -equivariant refinements of THH and the K -theory of endomorphisms whose fixed pointsfor varying r are linked together. This requires some generalizations of the constructionsand theorems of the previous three sections. For a short review of equivariant spectraand the notation we are using see §2.4. We then define the equivariant Dennis trace andgive a description of its effect on π in terms of the bicategorical tracetr( f r ◦ · · · ◦ f )of a composite twisted endomorphism (Proposition 7.18).7.1. r -fold endomorphisms and THH ( r ) .Definition 7.1. Given a spectral Waldhausen category C , let C × r denote the r -fold prod-uct of the base category, with Waldhausen structure determined coordinate-wise. Let ρ denote the rotation functor ρ : ( c , . . ., c r ) ( c , . . ., c r , c ).Given a twisting L C / D R of C , let R r ρ = R r ◦ ρ = ρ ◦ R r : C × r → D × r denote the exact functor R r ρ ( c , . . ., c r ) = ( R ( c ), . . ., R ( c r ), R ( c )). Definition 7.2.
Define the r -fold twisted endomorphism category of C / D L R byEnd ( r ) ¡ C / D L R ¢ : = End ³ ¡ C × r / D × r ¢ L r R r ρ ´ .There is an exact functor ∆ r : End( C / D L R ) → End ( r ) ( C / D L R )taking ( a , f ) to (( a , . . ., a ), ( f , . . ., f )); we refer to this as the duplication functor .An object of this category consists of an r -tuple of objects ( a , . . ., a r ) in C and an r -tuple of morphisms in D ¡ f : L ( a ) −→ R ( a ), . . ., f r − : L ( a r − ) −→ R ( a r ), f r : L ( a r ) −→ R ( a ) ¢ .See Figure 7.3a. A morphism ( a i , f i ) −→ ( b i , g i ) is an r -tuple of morphisms ( t i : a i −→ b i )such that R ( t i + ) ◦ f i = g i ◦ L ( t i ) (indices taken mod r ).See Figure 7.3b. Definition 7.4.
Suppose C is a pointwise cofibrant spectral category and X is a pointwisecofibrant ( C , C ) bimodule. For each r ≥
1, let C ∧ r be the spectral category with object set(ob C ) × r and mapping spectra C ∧ r (( a , . . ., a r ), ( b , . . ., b r )) = r ^ i = C ( a i , b i ).As in Definition 7.1 we let ρ : C ∧ r → C ∧ r , ρ ( c , . . ., c r ) = ( c , . . ., c r , c )be the spectral functor that rotates the smash product factors.Similarly, let X ∧ r denote the ( C ∧ r , C ∧ r )-bimodule whose value on each pair of r -tuplesis the evident r -fold smash product. Twisting on the right by ρ gives another ( C ∧ r , C ∧ r )-bimodule X ∧ r ρ . We define r -fold topological Hochschild homology by the formulaTHH ( r ) ( C ; X ) : = THH( C ∧ r ; X ∧ r ρ ). -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 31 L ( a ) f / / R ( a ) · · · L ( a r − ) f r − / / R ( a r ) L ( a r ) f r / / R ( a ) a B B B(cid:2)B(cid:2)B(cid:2) a \ \ \(cid:28) \(cid:28) \(cid:28) B B B(cid:2)B(cid:2)B(cid:2) a r − \ \ \(cid:28) \(cid:28) \(cid:28) B B B(cid:2)B(cid:2)B(cid:2) a r \ \ \(cid:28) \(cid:28) \(cid:28) B B B(cid:2)B(cid:2)B(cid:2) a \ \ \(cid:28) \(cid:28) \(cid:28) ( A ) Objects of End ( r ) ¡ C / D L R ¢ a t (cid:15) (cid:15) % % %e%e%e a y y y9 y9 y9 y9 t (cid:15) (cid:15) · · · a rt r (cid:15) (cid:15) $ $ $d$d$d a y y y9 y9 y9 y9 t (cid:15) (cid:15) L ( a ) L ( t ) (cid:15) (cid:15) f / / R ( a ) R ( t ) (cid:15) (cid:15) · · · L ( a r ) L ( t r ) (cid:15) (cid:15) f r / / R ( a ) R ( t ) (cid:15) (cid:15) L ( b ) g / / R ( b ) · · · L ( b r ) g r / / R ( b ) b : : :z:z:z:z b d d d$ d$ d$ d$ · · · b r : : :z:z:z:z b . d d d$ d$ d$ d$ ( B ) Morphisms of End ( r ) ¡ C / D L R ¢ F IGURE X ∧ r ρ gives an isomorphism of bimodules ρ : X ∧ r ρ −→ X ∧ r ρ overthe map of spectral categories ρ : C ∧ r −→ C ∧ r . In other words, we get an action of thecyclic group C r on the pair consisting of the spectral category C ∧ r and the bimodule X ∧ r ρ .This makes r -fold THH into an orthogonal C r -spectrum. Proposition 7.5.
When C and X are pointwise cofibrant, THH ( r ) is cofibrant and theHill–Hopkins–Ravenel norm diagonal of Proposition 2.20 induces isomorphisms of or-thogonal C s -spectra Φ C r THH ( rs ) ( C ; X ) ∼= THH ( s ) ( C ; X ) for all r , s ≥ .Proof. For the cofibrancy statement it suffices to check that the latching maps of thesimplicial spectrum defining THH ( r ) are cofibrations of orthogonal C r -spectra. Theselatching maps are cofibrations since the r -fold smash power turns cofibrations into equi-variant cofibrations [Mal17a, 4.11].For the second part, we commute Φ C r with the realization and apply the norm diago-nal D r at every simplicial level: Φ C r ¯¯¯¯¯¯ [ k ] _ ( a i ),...,( a ki ) C ∧ rs (( a i ),( a i )) ∧ ··· ∧ C ∧ rs (( a k − i ),( a ki )) ∧ X ∧ rs ρ (( a ki ),( a i )) ¯¯¯¯¯¯ ∼= ¯¯¯¯¯¯ [ k ] _ ( b i ),...,( b ki ) Φ C r ³ C ∧ rs (( b i ) × r ,( b i ) × r ) ∧ ··· ∧ C ∧ rs (( b k − i ) × r ,( b ki ) × r ) ∧ X ∧ rs ρ (( b ki ) × r ,( b i ) × r ) ´¯¯¯¯¯¯ ∼= ¯¯¯¯¯¯ [ k ] _ ( b i ),...,( b ki ) C ∧ s (( b i ),( b i )) ∧ ··· ∧ C ∧ s (( b k − i ),( b ki )) ∧ X ∧ s ρ (( b ki ),( b i )) ¯¯¯¯¯¯ . Here ( a ji ) = ( a j , a j , . . ., a jrs ) ranges over rs -tuples of objects of C , ( b ji ) = ( b j , b j , . . ., b js )ranges over s -tuples of objects of C , and( b ji ) × r = ( b j , . . ., b js , . . ., b j , . . ., b js ) is the rs -tuple obtained by duplicating an s -tuple r times. It remains to check that thediagonal respects the faces and degeneracy maps, and the C s -action. For degeneraciesand every face but the first, this follows by naturality of the diagonal. For the first facethis follows from [Mal17a, 3.26], and for the C s -action it follows from [Mal17a, 3.27]. Inother words, both are consequences of the rigidity theorem [Mal17a, 1.2]. (cid:3) Remark . When X = C , the spectrum THH ( r ) ( C ; C ) is isomorphic to THH( C ) using r -foldsubdivision, and this is the cyclotomic structure constructed in [Mal17a, 4.6]. It is equiv-alent to Bökstedt’s cyclotomic structure on THH( C ) by the main result of [DMP + r -fold topologicalHochschild homology spectrum we get the construction illustrated in Figure 7.7. CCCC C C XXXX X X F IGURE ( r ) ( C ; X ) Proposition 7.8.
After forgetting the C r -action, there is a natural equivalence THH ( r ) ( C ; X ) ≃ THH( C ; X ⊙ · · · ⊙ X ) with r copies of X on the right.Proof. The spectrum THH( C ; X ⊙ r ) is built using r bar constructions, the final one beingthe cyclic bar construction, so it is the realization of an r -fold multisimplicial spectrum.Taking the diagonal of this multisimplicial spectrum, we identify the resulting simplicialspectrum with the one for THH ( r ) ( C ; X ) by regrouping copies of C and X . Note that theuse of the cyclic action ρ in the definition of THH ( r ) is essential to this argument. (cid:3) Remark . As a consequence of Corollary 4.20 and Propositions 2.19, 7.5 and 7.8, theconstruction THH ( r ) ( C ; D L R ) sends Morita equivalences in the C variable and Dwyer–Kan embeddings in the D variable to equivalences of C r -spectra. Consequently, the mapTHH ( r ) ( A ; M ) ∼ −→ THH ( r ) ( P A ; M A M )induced by the Morita equivalence A −→ P A (Example 4.16) is an equivalence of C r -spectra. Theorem 7.10 (Additivity of THH ( r ) ) . Let C be a spectral Waldhausen category. Thenthe maps ι i ,..., i n defined above Theorem 5.9 induce an equivalence of C r -spectra _ i ,..., i n ≤ i j ≤ k j THH ( r ) ( C ) ∼ −→ THH ( r ) ( w k S ( n ) k ,..., k n C ). For any twisting L , R : C ⇒ D , the maps ι i ,..., i n also induce an equivalence of C r -spectra _ i ,..., i n ≤ i j ≤ k j THH ( r ) ( C ; D L R ) ∼ −→ THH ( r ) ( w k S ( n ) k ,..., k n C ; ( w k S ( n ) k ,..., k n D ) L R ). -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 33 ( A ) ( B ) F IGURE c r L ( c r ) f r R ( c ) c L ( c ) f R ( c ) c L ( c ) f R ( c ) c C L D R C L D R C L D R C F IGURE
Proof.
By Propositions 2.19, 7.5 and 7.8, it suffices to check that the corresponding non-equivariant map _ i ,..., i n ≤ i j ≤ k j THH( C ; D L R ⊙ r ) ∼ −→ THH( w k S ( n ) k ,..., k n C ; ( w k S ( n ) k ,..., k n D ) L R ⊙ r ).is an equivalence. This is proven by the same induction as in Theorem 5.9. The onlydifference is that in the inductive step (proof of Proposition 5.10) we apply the map f atotal of r times instead of just once. (cid:3) Definition of the equivariant Dennis trace.
We next define the equivariantrefinement of the Dennis trace.
Remark . If C is a spectral Waldhausen category, then ( C ∧ r , C × r ) fails to be a spectralWaldhausen category because it does not satisfy the pushout axiom. Therefore the equi-variant Dennis trace is not the twisted Dennis trace applied to C ∧ r . Instead, the smashpowers have to occur on the outside of the S • construction. Definition 7.13.
Let C / D L R be a twisting of a spectral Waldhausen category C . Foreach r ≥ C r -equivariant map(7.14) Σ ∞ ob End ( r ) ¡ ( C / D ) L R ¢ −→ _ c ,..., c r ∈ ob C r ^ i = D ( L ( c i ), R ( c i + )) −→ THH ( r ) ( C ; D L R ).See Figure 7.15. This is natural with respect to morphisms of twisted spectral Wald- hausen categories, so we apply it to the multisimplicial spectral Waldhausen category w • S ( n ) • C twisted by w • S ( n ) • D , and get a zig-zag of multisimplicial orthogonal spectra Σ ∞ ob End ( r ) ³ ( w • S ( n ) • C / w • S ( n ) • D ) L R ´ −→ THH ( r ) ³ w • S ( n ) • C ; ( w • S ( n ) • D ) L R ´ ≃ ←− ( S • ) ∧ n ∧ THH ( r ) ¡ C ; D L R ¢ ,the second map coming from Theorem 7.10. The same argument as in Lemma 6.13shows that this is a zig-zag of Σ ∆ -diagrams of simplicial orthogonal C r -spectra, hencewe can take their geometric realization and get a zig-zag of C r -equivariant bispectra.We describe these in detail in [CLM + , Appendix A]. Applying left-derived prolongation([CLM + , Proposition A.7]), and using the canonical identifications of object setsob w • S ( n ) • End ( r ) ¡ C / D L R ¢ = ob End ( r ) ³ ( w • S ( n ) • C / w • S ( n ) • D ) L R ´ ,we get a zig-zag of orthogonal C r -spectra(7.16) P K End ( r ) ¡ C / D L R ¢ −→ ¯¯¯ THH ( r ) ¡ w • S ∗• C ; ( w • S ∗• D ) L R ¢¯¯¯ ≃ ←− THH ( r ) ¡ C ; D L R ¢ .The r -fold Dennis trace map trc ( r ) : K End ( r ) ¡ C / D L R ¢ −→ THH ( r ) ¡ C ; D L R ¢ is the map represented by this zig-zag in the stable homotopy category of orthogonal C r -spectra.The case r = (1) = trc : Σ ∞ K (End( L C / D R )) −→ THH( C ; D L R )from Definition 6.16.The following is an extension of Proposition 6.24:
Lemma 7.17.
Let ( f i : L ( a i ) −→ R ( a i + )) be an object of the r-fold twisted endomor-phism category End ( r ) ¡ C / D L R ¢ , where we take indices modulo r. The image of the class [ f , . . ., f r ] ∈ K End ( r ) ¡ C / D L R ¢ under the r-fold Dennis trace map trc ( r ) : π K End ( r ) ¡ C / D L R ¢ −→ π THH ( r ) ( C / D L R ) is the composite S V [ f i ] −−−→ r ^ i = D ( L ( a i ), R ( a i + )) −→ _ ( c ,..., c r ) ∈ C r r ^ i = D ( L ( c i ), R ( c i + )) -skeleton −−−−−−−→ THH ( r ) ( C / D L R ) which includes the homotopy classes of the f i as -simplices in the cyclic bar construction. We can use this result to identify the image under the trace of certain classes in the K -theory of endomorphisms. Let A be a ring spectrum and let M be an ( A , A )-bimodule.Given a collection P , . . ., P r of perfect left A -modules and A -module maps f i : P i −→ M ∧ A P i + ,where the indices are taken modulo r , we let f r ◦ · · · ◦ f denote the composite map P f −→ M ∧ A P ∧ f −−−−→ M ∧ A M ∧ A P ∧ f −−−−→ · · · id ∧ f r −−−−→ M ∧ A r ∧ A P .The bicategorical trace of f r ◦ · · · ◦ f is a map of spectratr( f r ◦ · · · ◦ f ) : S ∼= THH( S ) −→ THH( A ; M ∧ A r ).On the other hand, the collection ( f , . . ., f r ) is an object of the r -fold twisted endomor-phism category End ( r ) ( P / M −∧ A M ), and thus determines a class [ f , . . ., f r ] in K (End ( r ) ( P / M M )). -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 35 Proposition 7.18.
The image of the class [ f , . . ., f r ] ∈ K (End ( r ) ( P / M M )) under the com-posite map (7.19) K (End ( r ) ( P / M M )) THH ( r ) ( P ; M M ) THH ( r ) ( A ; M ) THH( A ; M ∧ A r ) trc ( r ) ≃ ≃ is the homotopy class of the trace of the composite [tr( f r ◦ · · · ◦ f )] ∈ π THH( A ; M ∧ A r ) .Proof. The homotopy class of f r ◦ · · · ◦ f is encoded by the map of spectra S ∼= S ∧ r V ri = [ f i ] −−−−−→ r ^ i = M ( P i , M ∧ A P i + ) ◦ −→ M ( P , M ∧ A r ∧ A P ).This fits into the following commutative diagram of inclusions and composition maps. (7.20) S r ^ i = M ( P i , M ∧ A P i + ) _ ( Q ,..., Q r ) ∈ ob P r M ( Q , M ∧ A Q ) ∧ ··· ∧ M ( Q r , M ∧ A Q ) _ Q _ ( Q ,..., Q r ) M ( Q , M ∧ A Q ) ∧ ··· ∧ M ( Q r , M ∧ A Q ) M ( P , M ∧ A r ∧ A P ) _ Q ∈ ob P M ( Q , M ∧ A r ∧ A Q ) V ri = [ f i ][ f ◦···◦ f r ] ◦ ◦ The right-hand column of the previous diagram is the 0-skeleton of the left-hand columnof the next diagram:(7.21) THH ( r ) ( P ; M M ) THH ( r ) ( A ; M )THH( P ; M ⊙ rM ) THH( A ; M ⊙ r )THH( P ; M M ∧ A r ) THH( A ; M ∧ A r ). ∼= ≃ ∼=◦ ≃ ≃≃ Here the horizontal arrows come from the Morita equivalence A → P and the map ofbimodules M → M M from Example 4.5. The upper vertical maps on the left and rightare the unwinding equivalence from Proposition 7.8 and the top region commutes bynaturality of this equivalence. We use the notation M ⊙− to denote the bar construction,to distinguish it from the strict smash product M ∧ A − . The lower-right vertical mapcollapses this bar construction, and the bottom region commutes using ([CLM + , Lemma3.16]).The lower-left vertical map of (7.21) arises by applying THH to the morphism of ( P , P )-bimodules M ⊙ rM −→ M M ∧ A r defined by iterating the composition operation M ( P , M ∧ A i ∧ A − ) ⊙ M ( − , M ∧ A P ′ ) id ∧− −→ M ( P , M ∧ A i ∧ A − ) ⊙ M ( M ∧ A i ∧ A − , M ∧ A ( i + ∧ A P ′ ) ◦ −→ M ( P , M ∧ A ( i + ∧ A P ′ )for i =
1, . . ., r −
1. The commutativity of the bottom region of (7.21) is the fact that whenwe take P = P ′ = A , the resulting composite map of ( A , A )-bimodule spectra M ∧ A r −→ M ( A , M ∧ A A ) ⊙ r ◦ −→ M ( A , M ∧ A r ∧ A A )is adjoint to the identity of M ∧ A r . Paste the diagrams (7.20) and (7.21) together by including the 0-skeleta into the THHterms. The composite along the top and right of the resulting diagram agrees with thecomposite in the statement of the proposition, by Lemma 7.17. The bottom composite isthe inclusion of the map f r ◦· · ·◦ f into the cyclic bar construction for THH( P ; M M ∧ A r ), fol-lowed by the Morita equivalence to THH( A ; M ∧ A r ). By [CP19, 7.4], this is the homotopyclass of the bicategorical trace [tr( f r ◦ · · · ◦ f )], and thus the proof is complete. (cid:3)
8. T
HE TRACE TO TOPOLOGICAL RESTRICTION HOMOLOGY
Now that we have constructed an equivariant refinement of the Dennis trace, wedistill out of it a trace from the K -theory of endomorphisms to topological restrictionhomology which we call the TR-trace. We then define an analog of the ghost coordinatesof Witt vectors for TR: the ghost maps g n : TR X • −→ X n . Finally, we prove that applyingthe n -th ghost map to the TR-trace encodes the trace tr( f ◦ n ) of the n -fold iterate of a self-map (Theorem 8.21).8.1. Restriction Systems.
We begin by formalizing the sense in which the r -fold Den-nis traces trc ( r ) fit together as r varies. The situation is a bit subtle because on the K -theory side they are related by the categorical fixed points, and on the THH side theyare related by the geometric fixed points. Definition 8.1.
For each m , n ≥
1, let Θ n be a functor from C mn -spectra to C m -spectrasuch that for all r , s ≥ Θ rs → Θ s ◦ Θ r .An Θ ∗ -pre-restriction system consists of the following data: • A sequence of spectra { X n } ∞ n = together with an action of C n on X n for all n . • A C s -equivariant map c r : Θ r ( X rs ) → X s for all r , s ≥ Θ rs ( X rst ) (cid:15) (cid:15) c rs / / X t Θ s ( Θ r ( X rst )) Θ s ( c r ) / / Θ s ( X st ) c s O O commute for all r , s , t ≥ Θ n is the categorical fixed points functor ( − ) C n and each c r is an isomorphism,this is called a naive restriction system . When Θ n is the geometric fixed points functor Φ C n , and each c r induces a weak equivalence out of the derived geometric fixed points,this is a genuine restriction system .A morphism of restriction systems consists of equivariant maps X n −→ Y n commutingwith the maps c r . Remark . Our definition of restriction systems recalls the structures defining a p -cyclotomic spectrum; however, instead of working with the pro-group Z p we are insteadworking with the pro-group b Z . Restriction systems should therefore be closely related topro-spectra; see, for example, [Fau08]. Example 8.3.
The definition of naive restriction system works equally well for the cat-egory of spaces, so let us consider that case first. Let X be a space equipped with aself-map f : X → X . There is a naive restriction system whose n th term is the twistedfree loop space of the Fuller construction L Ψ n ( f ) X n , consisting of all n -tuples of points x , . . ., x n and paths from f ( x i ) to x i + (indices modulo n ) [MP18b, KW10]. The casewhere all of the paths are constant gives a naive restriction system where the n -th levelis the n -periodic points Fix( f ◦ n ). In the case where f = id, the system assigns to each -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 37 n ≥ L X = Map( S , X ) with C n acting by rotating the loops. Thestructure maps are the n -power maps ( L X ) C n ∼= L X . Example 8.4. If T is a cyclotomic spectrum in the sense of [ABG +
18, BM15], there is agenuine restriction system where the n th term is T with the generator of C n acting by e π i / n ∈ S . Example 8.5.
We give a couple of examples of ways to go between different kinds ofrestriction system structures. If X • is a naive restriction system of based spaces, takingsuspension spectra Σ ∞ X • gives a genuine restriction system. This uses the isomorphism Φ C r Σ ∞ X ∼= Σ ∞ ( X C r ), and the fact that in this case the geometric fixed points agree withthe left-derived geometric fixed points.Now suppose instead that X • is a genuine restriction system of fibrant orthogonalspectra. For any G -spectrum Y there is a canonical map κ : Y G → Φ G Y from the cate-gorical fixed points to the geometric fixed points. There are therefore maps γ r : X C rs rst κ Cs −→ ( Φ C r X rst ) C s c Csr −→ X C s st .These maps make X • into a ( − ) C n -pre-restriction system.The key examples for the purposes of the current discussion are the following: Example 8.6.
There is a naive restriction system whose n -th level is the K -theory K (End ( n ) ( C / D L R )) of the n -fold endomorphism category from Definition 7.1. The struc-ture maps c r identify the fixed points of such a category with the same category for asmaller value of n : K ³ End ( rs ) ¡ C / D L R ¢´ C r ∼= K ³ End ( rs ) ¡ C / D L R ¢ C r ´ ∼= K End ( s ) ¡ C / D L R ¢ .In other words, an rs -tuple of objects and morphisms that is strictly preserved by the C r -action must be an s -tuple of objects and morphisms that are repeated r times. Example 8.7.
When C is a pointwise cofibrant spectral category and X is a pointwisecofibrant bimodule, the isomorphisms from Proposition 7.5 make THH • ( C ; X ) into a gen-uine restriction system. The proof is essentially that of [Mal17a, 4.6], but simpler be-cause there are no subdivisions.The r -fold Dennis trace map from Definition 7.13 assembles into a map of restrictionsystems. Since the Dennis trace is a zig-zag of bispectra, the most natural statementto make is that it defines a zig-zag of restriction systems of bispectra. The details ofthis notion are in Appendix A—the important thing to know is that the geometric fixedpoints of a bispectrum are taken at each symmetric spectrum level separately, so that inthe symmetric spectrum direction they behave like categorical fixed points, and in theorthogonal direction they behave like geometric fixed points. It is this convention thatrectifies the apparent disparity between K End ( r ) and THH ( r ) .We can suspend the naive restriction system of Example 8.6 and get a genuine re-striction system of bispectra Σ ∞ K ¡ End • ¡ C / D L R ¢¢ , as in Example 8.5. Similarly, we cansuspend the genuine restriction system of Example 8.7 and get a genuine restriction sys-tem of bispectra Σ ∞ THH • ¡ C ; D L R ¢ . See Examples A.2 and A.3 for additional discussion. Theorem 8.8.
The r-fold Dennis trace for varying r ≥ • : Σ ∞ K ¡ End • ¡ C / D L R ¢¢ −→ Σ ∞ THH • ¡ C ; D L R ¢ together define a morphism of genuine restriction systems of bispectra. By Propositions A.1 and A.6, we can therefore make these restriction systems cofi-brant and then prolong them back to orthogonal spectra, giving a map in the homotopycategory of genuine restriction systems of orthogonal spectra P Σ ∞ K ¡ End • ¡ C / D L R ¢¢ −→ THH • ¡ C ; D L R ¢ . Proof.
Apply Example 8.7 to the middle termTHH ( r ) ³ w • S ( n ) • C ; ( w • S ( n ) • D ) L R ´ of the zig-zag that defines the r -fold Dennis trace (7.16). By the naturality of the con-struction, this gives a Σ ∆ -diagram of restriction systems of simplicial orthogonal C r -spectra. Concretely, such an object consists of orthogonal spectra X k , k ,..., k n , r with thestructure of a Σ ∆ -diagram in the first ( n +
1) indices, and of a genuine restriction systemin the last index, that commute with each other. By naturality, the maps of the additiv-ity theorem for THH ( r ) (Theorem 7.10) give a map of Σ ∆ -diagrams of restriction systemsof simplicial orthogonal C r -spectra.We next check that the inclusion of K -theory (7.14) also gives a map of Σ ∆ -diagramsof restriction systems of simplicial orthogonal C r -spectra. We already know it commuteswith most of this structure by the argument before Definition 7.13; the only new thingto check is agreement with the maps of the restriction system. By Remark 2.21, we canrewrite the maps of the restriction system on the left using the HHR norm diagonal, andthen the inclusion of endomorphisms commutes with the restriction system structure,simply because the norm diagonal is natural: Φ C r Σ ∞ ob End ( rs ) ¡ ( C / D ) L R ¢ / / Φ C r _ a ,..., a rs ∈ ob C rs ^ i = D ( L ( a i ), R ( a i + )) Σ ∞ ob End ( s ) ¡ ( C / D ) L R ¢ D r O O / / _ b ,..., b s ∈ ob C s ^ i = D ( L ( b i ), R ( b i + )). D r O O Each of the desired two maps is now a map of systems of spectra X k , k ,..., k n , r withboth a Σ ∆ -action and with restriction maps. We re-interpret these as Σ ∆ -diagrams ofrestriction systems and take their realization to get symmetric spectrum objects in pre-restriction systems. We identify these with pre-restriction systems of bispectra by notingthat the geometric fixed point functor for bispectra is (uniquely) isomorphic to the func-tor that takes the geometric fixed points at each symmetric spectrum level separately([CLM + , Definition A.9]).It remains to check that we actually have restriction systems, in other words that thegeometric fixed points agree with the left-derived geometric fixed points. We alreadyknow this for K -theory and THH but not for the middle term of our zig-zag. To verifythis condition, it is enough if each of the C r -bispectra Y • , • is a cofibrant orthogonal C r -spectrum at each symmetric spectrum level Y n , • . In other words, we must show thatthe realization in the k through k n directions, | X • , • ,..., • , r | , is a cofibrant orthogonal C r -spectrum.To accomplish this we use the model structure on restriction systems of orthogonal C r -spectra from Proposition A.6 and then take cofibrant replacement in Σ ∆ -diagramsof such objects by [CLM + , Theorem 6.4]. Then for fixed n , each system X k , k ,..., k n , r is Reedy cofibrant, meaning each latching map is a cofibration of restriction systems. -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 39 Therefore for each fixed value of r the latching map is a cofibration of orthogonal C r -spectra. Therefore the realization is a cofibrant orthogonal C r -spectrum. Therefore,after cofibrant replacement we get a zig-zag of maps of restriction systems of bispectra.Note that after this replacement, the backwards map of the zig-zag is an equivalenceof bispectra at level 1 of the restriction system. By Proposition 2.19, it is therefore anequivalence of C r -bispectra at level r for every r ≥
1. This finishes the proof. (cid:3)
Topological restriction homology.Definition 8.9.
For any Θ ∗ -pre-restriction system, the spectra Θ n ( X n ) assemble into adiagram indexed by the category I with one object for each n ≥
1, one morphism n → m when m | n , and no other morphisms (as in [Mad94, §2.5]). For any Θ ∗ -pre-restrictionsystem, the spaces Θ n ( X n ) assemble into a diagram indexed by the category I .Let X • be a genuine restriction system. Define maps(8.10) X C rs rs ∼= ( X C r rs ) C s κ Cs / / ( Φ C r X rs ) C s ( γ r ) Cs / / X C s s by first composing the canonical map κ from categorical fixed points to geometric fixedpoints and the structure map γ r . Then apply categorical C s -fixed points. As above,this makes the spectra X C n n into a diagram indexed by I . We define the underivedtopological restriction homology of X • byTR un ( X • ) = lim I X C n n .If in addition each X r is fibrant as an orthogonal C r -spectrum, we define the topo-logical restriction homology of X • as the homotopy limitTR( X • ) = holim I X C n n .To define TR for an arbitrary genuine restriction system, we first take a fibrant replace-ment in the model structure from Proposition A.6, and then take the homotopy limit asabove. There is a canonical map TR un → TR that takes fibrant replacement and passesto the homotopy limit.
Definition 8.11.
For any pointwise cofibrant spectral category C and pointwise cofibrant( C , C ) bimodule X , we write TR( C ; X ) for the TR of the restriction system THH • ( C ; X ) fromExample 8.7.When C = A is a ring spectrum and X = A , this is the classical definition of topologicalrestriction homology. On the other hand, if we take a general ( A , A )-bimodule spectrum M , this is a spectral version of Lindenstrauss–McCarthy’s W ( A ; M ) [LM12].Before defining the TR-trace we need one more technical result. Lemma 8.12 (Example A.2) . Let X • be a naive restriction system of symmetric spectra.Then P Σ ∞ X • is a genuine restriction system of orthogonal spectra and there is an isomor-phism TR un ( P Σ ∞ X • ) ∼= P Σ ∞ X ∼= P X .We are now ready to define the TR-trace. Definition 8.13.
The TR -trace is the map trc : K (End( C / D L R )) → TR( C ; D L R ) definedas the composition K (End( C / D L R )) ∼= TR un ( P Σ ∞ K (End • ( C / D L R ))) −→ TR( P Σ ∞ K (End • ( C / D L R ))) −→ TR(THH • ( C ; D L R )) = TR( C ; D L R ).(8.14)
Example 8.15.
Taking C = P A for a ring spectrum A , and twisting by an ( A , A )-bimodule M , the TR-trace gives a map(8.16) trc : K ( A ; M ) = K End( P / M M ) −→ TR ( P ; M M ) ∼ ←− TR( A ; M ).As before, the TR-trace is identically zero on the zero endomorphisms, so induces a mapout of cyclic K -theory(8.17) trc : K cyc ( A ; M ) = e K ( A ; M ) −→ TR( A ; M ). Remark . We briefly discuss two compatibility statements with the existing litera-ture. One is that when M = A and we restrict to identity morphisms, we recover theusual trace K ( P A ) → TR( P A ) as in e.g. [BHM93]. Recall that the more common def-inition arises by using the fact that the inclusion of identity morphisms into THH( C )lands in the categorical fixed points THH( C ) S , and then constructing a natural mapTHH( C ) S → TR( C ) out of r -fold subdivision and the restriction maps in equivariant sta-ble homotopy theory. However, the r -fold subdivision applied to the inclusion of identitymorphisms (6.8) agrees with the equivariant inclusion of identity morphisms in (7.14),so our construction produces the same map to TR.The other statement is that our trace agrees with the trace to W ( A ; M ) as defined byLindenstrauss and McCarthy for discrete rings [LM12]. The comparison can be sketchedjust as in Remark 6.22, only the equivalences are C r -equivariant and we compare the r -fold inclusion of endomorphisms map (7.14) with the one in [LM12, 9.1].8.3. The ghost map.Definition 8.19.
Let X • be a genuine restriction system. The ghost map g = ( g n ) : TR( X • ) → Y n ≥ X n is defined by the composites g n : TR( X • ) X C n n X n , F n where F n denotes the inclusion of fixed points.The ghost map is defined in the same way for underived TR. It is natural with respectto maps of restriction systems, and with respect to the inclusion of underived TR intoTR, and this naturality makes it easy to compute. Proposition 8.20.
Let C be a spectral Waldhausen category and let L C / D R be a twistingof C . Then the following diagram commutesK End ¡ C / D L R ¢ ∆ n (cid:15) (cid:15) trc / / TR ¡ C ; D L R ¢ g n (cid:15) (cid:15) K End ( n ) ¡ C / D L R ¢ trc ( n ) / / THH ( n ) ¡ C ; D L R ¢ , where ∆ n is the duplication functor from Definition 7.2. Setting n = , we conclude thatthe TR -trace is a lift of the Dennis trace along the first ghost map g :K End ¡ C / D L R ¢ trc (1) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ trc / / TR ¡ C ; D L R ¢ g (cid:15) (cid:15) THH ¡ C ; D L R ¢ . -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 41 Proof.
By Lemma 8.12, the underived TR of the genuine restriction system P Σ ∞ K End • ¡ C / D L R ¢ is the K -theory of endomorphisms K End ¡ C / D L R ¢ . Using the naturality of the ghostmap, it suffices to prove that under this identification, the n th ghost map g n : TR un P Σ ∞ K End • ¡ C / D L R ¢ −→ K End ( n ) ¡ C / D L R ¢ agrees with the map induced by ∆ n .The n th ghost map for the restriction system applies the inverse of the structuralisomorphisms γ n (see Example A.2) to get to the C n -fixed points of the n th term, thenincludes the C n -fixed points into the entire K -theory spectrum K End (1) ¡ C / D L R ¢ γ − n −→ K ³ End ( n ) ¡ C / D L R ¢´ C n F n −→ K End ( n ) ¡ C / D L R ¢ .It follows from the definition of γ n that this composite is the map induced by the dupli-cation functor ∆ n . (cid:3) Using the ghost map, we can extend the result of Proposition 7.18.
Theorem 8.21.
Let A be a ring spectrum, let P be a perfect A-module, and let M be an ( A , A ) -bimodule. The image of the class [ f ] ∈ K ( A ; M ) determined by a twisted endomor-phism f : P → M ∧ A P under the composite (8.22) K ( A ; M ) TR( A ; M ) THH ( n ) ( A ; M ) THH( A ; M ∧ A n ) trc g n ≃ is the homotopy class of the trace of the iterate [tr( f ◦ n )] ∈ π THH( A ; M ∧ A n ) .Proof. By naturality of the ghost map, we may apply g n before simplifying from P to A . This gives the top route in the commutative square of Proposition 8.20 followed bythe equivalence THH ( n ) ( P ; M M ) ≃ THH( A ; M ∧ A n ). The composite then agrees with thebottom route of Proposition 8.20 composed with this equivalence, and thus, by Proposi-tion 7.18 with each f i = f , takes [ f ] to [tr( f ◦ n )] ∈ π THH( A ; M ∧ A n ). (cid:3)
9. C
HARACTERISTIC POLYNOMIALS , ZETA FUNCTIONS , AND THE R EIDEMEISTERTRACE
In this section we explain how the characteristic polynomial, the Reidemeister trace,and the Lefschetz zeta function are all encoded by the TR-trace. The first result is thatwhen A is a commutative Eilenberg–Maclane spectrum, the TR-trace K End( A ) → π TR( A ) ∼= W ( A ) ∼= (1 + tA [[ t ]]) × takes each endomorphism [ f ] ∈ K End( A ) to its characteristic polynomial (Theorem 9.9).Here W ( A ) ∼= (1 + tA [[ t ]]) × is the ring of big Witt vectors of A , and the isomorphism π TR( A ) ∼= W ( A ) is a result of Hesselholt and Madsen [HM97], see also [Hes97, DKNP20].As a result, the TR-trace of this paper is a generalization of the characteristic polynomialmap K End( A ) → (1 + tA [[ t ]]) × studied by Almkvist and others [Alm74].The second result is that when A = Σ ∞+ Ω X is a spherical group ring with X finitelydominated and path-connected, each based map f : X → X defines a class[ f ] ∈ K ( Σ ∞+ Ω X ; Σ ∞+ Ω f X )whose image in TR records the Reidemeister traces R ( f n ) for all n ≥
1. Using [MP18b],this implies that the trace to TR takes [ f ] to its Fuller trace R ( Ψ n f ) C n for all n ≥ Z ) ∼= W ( Z ) then this classbecomes the Lefschetz zeta function of f (Theorem 9.22). In each of the above two cases there is a splitting of π TR into an infinite product,but the splittings arise for very different reasons. We refer to the splitting of TR( Σ ∞+ Ω X )as tom Dieck coordinates and the splitting of TR( H A ) for a commutative ring A as Witt coordinates . These should not be confused with each other, nor should they beconfused with the image under the ghost map, which we call ghost coordinates.
Whenwe apply the ring homomorphism S → Z to move between the above two examples, allthree of these coordinate systems come into play (Proposition 9.24). The distinctionbetween them is needed to fully comprehend how the Lefschetz zeta function of a map f : X → X is related to the Fuller trace R ( Ψ n f ) C n .9.1. The case of commutative Eilenberg–MacLane spectra.
The case of commu-tative Eilenberg–MacLane spectra is intimately tied with the Witt vectors. There is aconceptual reason for this: by Theorem 8.21, the TR-trace records the traces tr( f ◦ n ) ofthe iterates of an endomorphism f . The structure of the Witt vectors collates this infor-mation into a single class in π TR( A ) which corresponds to the characteristic polynomial χ f ( t ) = det(1 − t f ).In this section, A is a discrete commutative ring. We abuse notation and make theabbreviations TR( A ) = TR(
H A ), etc., so that we have no need to explicitly refer to theassociated Eilenberg–MacLane spectrum.We briefly recall the basic facts regarding the ring of (big) Witt vectors W ( A ) of A , referring the reader to [Alm74, Gra78, Hes03, Cam19] for more details. As a set, W ( A ) = A Z + consists of collections a = ( a n ) of ring elements a n ∈ A indexed by the posi-tive integers n ≥
1. We equip W ( A ) with the addition and multiplication uniquely deter-mined by the requirement that the ghost coordinates map(9.1) w = ( w n ) : W ( A ) −→ A Z + , w n ( a ) = X d | n da n / dd is a natural transformation of functors from commutative rings to commutative rings,where the ring structure on A Z + is defined componentwise. There is a natural isomor-phism of abelian groups(9.2) W ( A ) ∼= (1 + tA [[ t ]]) × defined by a Y n ≥ (1 − a n t n ),and the group of invertible power series can be given an additional binary operation forwhich this is an isomorphism of rings. The relevance of the ring structure on W ( A ) ∼= (1 + tA [[ t ]]) × , for our purposes, is that the characteristic polynomial map K (End( A )) χ −→ (1 + tA [[ t ]]) × [ f : P → P ] det(1 − t f )is a ring homomorphism. In fact: Theorem 9.3 ([Alm74]) . For any discrete commutative ring A, χ induces an injectivering homomorphism e K (End( A )) : = K (End( A )) / ι K ( A ) χ −→ (1 + tA [[ t ]]) × and the image consists of precisely those power series that are quotients of polynomials. Corollary 9.4.
A surjective ring homomorphism A → B induces a surjection e K (End( A )) → e K (End( B )).In order to state Hesselholt–Madsen’s isomorphism W ( A ) ∼= π TR( A ) in a useful way,we will also need the Witt vectors W 〈 n 〉 ( A ) = A 〈 n 〉 indexed on the truncation set 〈 n 〉 = { d ∈ Z + : d | n } . -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 43 The set W 〈 n 〉 ( A ) is made into a commutative ring by declaring that the ghost coordi-nates map w : W 〈 n 〉 ( A ) −→ A 〈 n 〉 , defined as in (9.1), is a ring homomorphism. The re-striction maps R n / d : W 〈 n 〉 ( A ) −→ W 〈 d 〉 ( A ), which forget the elements indexed by divisorsof n that do not divide d , are also ring homomorphisms. We make the identification W ( A ) ∼= lim n W 〈 n 〉 ( A ), where the limit is taken over the restriction maps.We now recall the result of Hesselholt–Madsen, expressed in terms of the restrictionsystem THH • ( A ) from §8.1. Theorem 9.5. [HM97, Add. 3.3]
There is a natural isomorphism of ringsI n : W 〈 n 〉 ( A ) ∼= −→ π THH ( n ) ( A ) C n defined by I n ( a ) = X d | n V d ( ∆ n / d ( a d )).Here ∆ r : A ∼= π THH( A ) −→ π THH ( r ) ( A ) C r is the duplication map induced by the map a a ∧ r on 0-skeleta, and V d : π THH ( r ) ( A ) C r −→ π THH ( dr ) ( A ) C dr is the Verschiebung map on THH, defined in terms of the equivariant transfer for thesubgroup C r < C dr . The isomorphism I respects the ghost coordinate maps on its domanand codomain, in the sense that g d ◦ I n = w d for every d | n , where g d : π THH ( n ) ( A ) C n R n / d −−−→ π THH ( d ) ( A ) C d F d −−→ π THH ( d ) ( A ) ∼= A is the d -th ghost coordinate map on π THH ( n ) ( A ) C n . Proof.
Since our conventions are different from those of Hesselholt–Madsen, it’s worthsaying something about the proof. The main point is that under the equivalenceTHH ( n ) ( A ) ≃ THH( A )of Proposition 7.8 (see also Remark 7.6), our definitions of the restriction map R n , interms of the restriction system structure map, and the Frobenius map F n , in termsof the inclusion of fixed-points, agree with Displays (1) and (16) of [HM97]. It followsthat on the n -th component THH ( n ) ( A ) C n of the restriction system, our ghost map g (seeDefinition 8.19) agrees with theirs (denoted by w ). The statement of the theorem thenfollows as in [HM97, Add. 3.3]. (cid:3) Lemma 9.6.
There is a natural isomorphism of rings I : W ( A ) ∼= −→ π TR( A ) that respectsthe ghost coordinate maps, meaning that g ◦ I = w.Proof. The map we want is essentially I = lim n I n , the limit over the restriction maps R of the isomorphisms I n . Technically, this produces an isomorphism to lim n π THH( A ) C n ,but it lifts to an isomorphism to π lim n THH( A ) C n because lim π THH( A ) C n =
0. Thislast claim follows from the surjectivity of R in the proof of [HM97, Prop 3.3], whichgeneralizes from the p -typical case by the discussion on [HM97, p. 55]. (cid:3) Let f : P → P be an endomorphism of a finitely generated projective A -module, withassociated K -theory class [ f ] ∈ K (End( A )). By Theorem 8.21, the image of [ f ] under theTR-trace and the ghost map is(tr( f ), tr( f ◦ ), tr( f ◦ ), . . .) ∈ Y n ≥ π THH ( n ) ( A ; A ) ∼= Y n ≥ A . As explained in [Hes03, Cam19], along the isomorphism W ( A ) ∼= (1 + tA [[ t ]]) × of (9.2),the ghost map of W ( A ) is identified with the negative logarithmic derivative(1 + tA [[ t ]]) × − t ddt log / / tA [[ t ]] ∼= Y n ≥ A . Lemma 9.7.
The element of Q n ≥ A given by the iterated traces (tr( f ◦ n )) is the negativelogarithmic derivative of the characteristic polynomial χ f ( t ) = det(id − t f ) ∈ (1 + tA [[ t ]]) × .In other words, the following diagram commutes.(9.8) e K (End( A )) trc v v ❧❧❧❧❧❧❧❧❧ χ ) ) ❙❙❙❙❙❙❙❙❙❙ π TR( A ) Q g n (cid:15) (cid:15) (1 + tA [[ t ]]) ×− t ddt log (cid:15) (cid:15) Q ∞ n = A o o / / tA [[ t ]] Proof.
When A is an algebraically closed field, this follows by induction on the eigenval-ues. It therefore holds for any integral domain, by passing to the algebraic closure of thefraction field. In particular, it holds for any polynomial ring over Z . If A is a generalcommutative ring, then there is a surjection from a polynomial ring to A . Using Corol-lary 9.4, the statement therefore holds for A as well. See [Cam19, 4.24] for a differentderivation. (cid:3) Theorem 9.9.
Let A be a commutative ring. Then the following triangle commutes. (9.10) e K (End( A )) trc w w ♣♣♣♣♣♣♣♣♣♣♣ χ ' ' PPPPPPPPPPPP π TR( A ) ∼= I − / / W ( A ) ∼= (9.2) / / (1 + tA [[ t ]]) × In other words, as invariants underneath the K -theory of endomorphisms, the trace to π TR( A ) is isomorphic to the characteristic polynomial.Proof. Pasting the diagrams in Displays (9.8) and (9.10) together gives the diagram e K (End( A )) trc w w ♣♣♣♣♣♣♣♣♣♣♣ χ ' ' PPPPPPPPPPPP π TR( A ) Q g n (cid:15) (cid:15) W ( A ) ∼= I o o Q w n w w ♣♣♣♣♣♣♣♣♣♣♣♣ ∼= / / (1 + tA [[ t ]]) ×− t ddt log (cid:15) (cid:15) Q ∞ n = A o o / / tA [[ t ]].The maps along the outside edge commute by the previous discussion, as does the trape-zoid and the small triangle on the left (Lemma 9.6). If A is torsion-free, the two verticalmaps are injective and therefore the triangle at the top commutes as well.To extend to the case where A is any commutative ring we use a standard trick (seee.g. [Gra78, p. 6]). Pick a surjective ring homomorphism A ′ → A with A ′ torsion-free, -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 45 and observe that the diagram is natural in ring homomorphisms. This gives a map fromthe diagram for A ′ to the diagram for A , that is surjective on the terms π TR( A ) ∼= W ( A ) ∼= (1 + tA [[ t ]]) × and on e K (End( A )) by Corollary 9.4. Therefore the desired triangle for A can be deducedfrom the same triangle for A ′ . (cid:3) Corollary 9.11.
Let A be a commutative ring. Then the TR-trace e K (End( A )) → TR( A ) isinjective on π . The case of spherical group rings and fixed-point theory.
As in the case ofEilenberg–MacLane spectra, the computation of π TR( A ) for A = Σ ∞+ G a spherical groupring hinges on the interplay between the ghost coordinates and a set of splitting coordi-nates for π TR( A ). However the splitting here arises for a very different reason, namelythe tom Dieck splitting from equivariant stable homotopy theory. Proposition 9.12.
If X • is a naive restriction system of spaces, then the TR of its suspen-sion genuine restriction system Σ ∞ X • from Example 8.5 has a tom Dieck splitting TR( Σ ∞ X • ) ≃ Y j ≥ ( Σ ∞ X j ) hC j . Proof.
We interpret the naive restriction system as a Z -space X with no free orbits, sothat X n = X n Z . Under the identification C n = Z / n Z , the generator of the cyclic group C n acts on X n = X n Z by 1 ∈ Z , and the generator of the subgroup C n / j < C n acts by j ∈ Z foreach positive divisor j | n .The proposition is a consequence of the classical tom Dieck splitting theorem, whichtells us that the derived fixed point spectrum of the suspension spectrum of X n splits asa wedge of homotopy orbits( Σ ∞ X n ) C n = ( Σ ∞ X n Z ) C n ≃ _ j | n Σ ∞ ( X j Z ) hC j = _ j | n Σ ∞ ( X j ) hC j .Here all of the fixed points labeled by cyclic groups C i are genuine fixed points, i.e.they are implicitly right-derived. We also need the standard fact that the restriction( Σ ∞ X n Z ) C n → ( Σ ∞ X k Z ) C k for k | n corresponds along this splitting to the map that re-stricts to the summands where j | k .These statements only apply in the homotopy category, so they do not directly implyanything about the homotopy limit defining TR. However, the map from j th summandis described more concretely as a transfer followed by a splitting of the restriction map: Σ ∞ ( X j Z ) hC j tr f −−→ ³ Σ ∞ X j Z ´ C j −→ ³¡ Σ ∞ X n Z ¢ C n / j ´ C j = ( Σ ∞ X n Z ) C n .Therefore, if we pick a representative for the splitting t n : ( Σ ∞ X n ) C n → ( Σ ∞ X n ) hC n foreach n ≥
1, the j th term of the splitting in the homotopy category is given by the formula( Σ ∞ X n ) C n = ³¡ Σ ∞ X n ¢ C n / j ´ C j κ −→ ¡ Σ ∞ X j ¢ C j t j −→ Σ ∞ ( X j ) hC j .This defines a map of homotopy limit systems from ( Σ ∞ X • ) C • to a product of homotopylimit systems over j ≥
1, the j th system having n th term Σ ∞ ( X j ) hC j when j | n and ∗ when j ∤ n , with all identity maps between them. On each term this map of homotopylimit systems is an equivalence by the above discussion, so it induces an equivalence ofhomotopy limits. This gives the desired tom Dieck splitting of TR. (cid:3) Remark . There is a generalization of the tom Dieck splitting theorem due to GaunceLewis [Lew00] that applies to the orthogonal C r -spectrum P Σ ∞ K ¡ End ( r ) ¡ C / D L R ¢¢ . Wecould use this together with the argument in Proposition 9.12 below to identify the mid-dle term of (8.14) as the infinite product(9.14) Y n ≥ K ³ End ( n ) ¡ C / D L R ¢´ hC n .The inclusion of underived TR is then just the first term in this product. Note that themap to the TR system described in Proposition 9.12 does not respect this splitting. Example 9.15.
Along the tom Dieck splitting, the ghost map is a map of products g : Y j ≥ ( Σ ∞ X j ) hC j −→ Y n ≥ Σ ∞ X n .To compute its n th coordinate we use the product system from the proof of Proposi-tion 9.12: TR( Σ ∞ X • ) ≃ (cid:15) (cid:15) / / ( Σ ∞ X n ) C n ≃ (cid:15) (cid:15) F / / Σ ∞ X n Y j ≥ ( Σ ∞ X j ) hC j / / Y j | n ( Σ ∞ X j ) hC j ♦♦♦♦♦ We then use [Mal17b, 4.4] to compute the dashed composite as the sum over all j | n ofthe transfers and inclusions(9.16) ( Σ ∞ X j ) hC j trf −→ Σ ∞ X j ∼= Σ ∞ ( X n ) C n / j −→ Σ ∞ X n .On π , the composite takes every path component of ( X j ) hC j to the weighted sum of itspreimage components in X j (weighted evenly so that the total weight is j ), then mapsforward to the corresponding components of X n . Here is a useful consequence of thisdescription in the case j = n . Proposition 9.17.
For a suspension spectrum restriction system, the ghost map on π g = ( g n ) : π TR( Σ ∞ X • ) −→ Y n ≥ π Σ ∞+ X n ∼= Y n ≥ H ( X n ) is injective. We now apply the tom Dieck splitting to TR of a spherical group ring. Let G be atopological group or grouplike topological monoid and write S [ G ] = Σ ∞+ G for its suspen-sion spectrum. Let A be a topological space with commuting left and right G -actions.Assume that G and A are cofibrant as topological spaces. Then the restriction systemTHH • ( S [ G ]; S [ A ]) from Example 8.7 is the suspension spectrum of a naive restrictionsystem whose n th level is the bar construction in unbased spaces B ( G × n ; A × n ρ ). Proposi-tion 9.12 therefore gives a splitting (where the C j denotes coinvariants)TR( S [ G ]; S [ A ]) ≃ Y j ≥ Σ ∞+ B ( G × j ; A × j ρ ) hC j , π TR( S [ G ]; S [ A ]) ∼= Y j ≥ HH ( Z [ π G ] ⊗ j ; Z [ π A ] ⊗ j ρ ) C j , π TR( S [ G ]) ∼= Y j ≥ HH ( Z [ π G ])For a based connected CW complex X , we take G = Ω X to be any well-based topologi-cal group modeling the loop space of X . Let f : X → X be a basepoint preserving self-map -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 47 of X . We let A = Ω f X be Ω X with the usual right action, and left action twisted by f ,i.e. given by the composite Ω X × Ω X Ω f × id −−−−→ Ω X × Ω X mult −−−→ Ω X .The tom Dieck coordinates can then be described as π TR( S [ Ω X ]; S [ Ω f X ]) ∼= Y j ≥ HH ( Z [ π X ]; Z [ π X ] f ◦ j ) f ∗ where ( − ) f ∗ denotes coinvariants under the action of f ∗ on each copy of π X .The above is true in general, but if X is finitely dominated, then S is perfect as a left S [ Ω X ]-module, and we can pick out a distinguished class[ f ] ∈ K (End( S [ Ω X ]; S [ Ω f X ])).It is the twisted endomorphism of ( S [ Ω X ], S )-bimodule spectra(9.18) S ≃ −→ S [ Ω f X ] ∧ S [ Ω X ] S that is homotopy inverse to the map that collapses the bar construction on the right backto S . This is the suspension of the canonical isomorphism ∗ Ω X ∼= −→ Ω X f ⊙ ∗ Ω X arising from the fact that f and the identity agree after composing with X → ∗ . By[Pon10], its bicategorical trace is the Reidemeister trace R ( f ) : S → THH( S [ Ω X ]; S [ Ω f X ]). Theorem 9.19.
For any finitely dominated space X and basepoint preserving self mapf : X → X , the image of the class [ f ] from (9.18) under the compositeK ( S [ Ω X ]; S [ Ω f X ]) trc / / π TR( S [ Ω X ]; S [ Ω f X ]) g / / Y n ≥ HH ( Z [ π X ]; Z [ π X ] f ◦ n ) is the Reidemeister series ( R ( f ), R ( f ◦ ), R ( f ◦ ), . . .) .Proof. By Theorem 8.21, the n th factor of this map is the trace of the composite S ∼= S [ Ω f X ] ∧ S [ Ω X ] S ∼= S [ Ω f X ] ∧ S [ Ω X ] S [ Ω f X ] ∧ S [ Ω X ] S ∼= · · · ∼= S [ Ω f X ] ∧ S [ Ω X ] ( n ) ∧ S [ Ω X ] S .Collapsing the copies of S [ Ω f X ] together gives the bimodule S [ Ω f ◦ n X ] and the twistedendomorphism (9.18) with f replaced by f ◦ n . Therefore, along the resulting isomorphism(9.20) THH( S [ Ω X ]; S [ Ω f X ] ∧ S [ Ω X ] n ) ∼= THH( S [ Ω X ]; S [ Ω f ◦ n X ])this trace is taken to the Reidemeister trace of f ◦ n . (cid:3) Remark . The TR-trace without coefficients A ( X ) ≃ K ( S [ Ω X ]) → TR( S [ Ω X ]) there-fore gives traces of the identity map, as in [Lyd95].The ring map π : S [ Ω X ] → S that collapses Ω X to a point and the corresponding bi-module map S [ Ω f X ] → S induce a map on topological restriction homologyTR( S [ Ω X ]; S [ Ω f X ]) π / / TR( S ).We may further compose with the ring map S → H Z to land in TR( Z ). Theorem 9.22.
The compositeK ( S [ Ω X ], S [ Ω f X ]) trc −−→ π TR( S [ Ω X ]; S [ Ω f X ]) π −→ π TR( S ) −→ π TR( Z ) ∼= (1 + t Z [[ t ]]) × maps the twisted module endomorphism [ f ] from (9.18) to the Lefschetz zeta function exp ÃZ t à ∞ X n = L ( f ◦ n ) t n !! = exp à ∞ X n = L ( f ◦ n ) n t n ! . Proof.
Naturality of the ghost map implies that the following diagram commutes, wherethe vertical maps are ghost maps:(9.23) π TR( S [ Ω X ]; S [ Ω f X ]) g (cid:15) (cid:15) / / π TR( S ) g (cid:15) (cid:15) / / (1 + Z [[ t ]]) ×− t ddt log (cid:15) (cid:15) Y n ≥ HH ( Z [ π X ]; Z [ π X ] f ◦ n ) / / Y n ≥ Z o o ∼= / / t Z [[ t ]]By Theorem 9.19, the class [ f ] in the upper-left goes to the Reidemeister traces R ( f ◦ n )in the lower-left. The augmentation sends these to the Lefschetz numbers L ( f ◦ n ) inthe lower-middle. This agrees with the image of the Lefschetz zeta function along thelogarithmic derivative. Since the logarithmic derivative is injective, we conclude thatthe image of [ f ] in the upper-right is the Lefschetz zeta function. (cid:3) In fact, the passage from S to Z in Theorem 9.22 has no effect on π TR:
Proposition 9.24.
The ring map S → H Z induces an isomorphism π TR( S ) ∼= π TR( Z ) ∼= (1 + t Z [[ t ]]) × . Proof.
Expanding the right-hand square of (9.23) using the tom Dieck splitting and Wittcoordinates, we get the commuting square Y j ≥ Z g (cid:15) (cid:15) / / Y i ≥ Z g (cid:15) (cid:15) ( b j ) ∞ j = ❴ (cid:15) (cid:15) ✤ / / ( a i ) ∞ i = ❴ (cid:15) (cid:15) Y n ≥ Z Y n ≥ Z ( w n ) ∞ n = ( w n ) ∞ n = .The ghost coordinates w n are given by the formulas X d | n db d = w n = X d | n da n / dd ,the first arising from Example 9.15 and the second from Display (9.1). It is an easyobservation that both of these maps are injective. Therefore the top horizontal map isinjective.To prove it is surjective, we reverse-engineer the second formula to write a n as n w n plus a rational polynomial in the a d for d | n , d < n . Inductively, this implies that a n is expressed as a rational polynomial in the b d for d | n , whose b n -term is n ( nb n ) = b n .Clearly changing the value of b n then allows us to attain any integer value of a n , so byinduction this collection of polynomials defines a surjective map to Q i ≥ Z . (cid:3) -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 49 Remark . The isomorphism π TR( S ) ∼= −→ π TR( Z ) is given from tom Dieck coordi-nates to Witt coordinates by the following polynomials in the first few degrees: a = b b = a a = b − b − b b = a + a − a a = b − b − b b = a + a − a The polynomials for a and b have many more terms, for instance a = b + µ b − b + b b − b b − b + b − b + b ¶ .Though it is not directly apparent from the formula, this polynomial is integer valuedon integer inputs.9.3. The relation to the free loop space and periodic-point theory.
For a space X and self-map f : X → X , let L f X denote the twisted free loop space L f X = { γ : [0, 1] → X | γ (0) = f ( γ (1)) } .In this final subsection we describe how the work of the previous subsection is related to[MP18b], which investigated the Reidemeister traces R ( f ◦ n ) as elements of π ( Σ ∞+ L f ◦ n X ) ∼= H ( L f ◦ n X ),rather than HH ( Z [ π X ]; Z [ π X ] f ◦ n ). When X is path-connected and f preserves a cho-sen basepoint, the two definitions of the Reidemeister trace agree along the equivalence(see [LM18, 6.5], [CP19, Cor A.14], [Mal19, 8.2.8])(9.26) THH( S [ Ω X ]; S [ Ω f X ]) ≃ Σ ∞+ L f X .We first lift this equivalence to an equivalence of restriction systems. Let Ψ n ( f ) = f × n ◦ ρ − denote the n th Fuller construction of f as in [KW10, MP18b]: X × · · · × X Ψ n ( f ) / / X × · · · × X ( x , x , . . ., x n ) ✤ / / ( f ( x n ), f ( x ), . . ., f ( x n − ))An element of the twisted free loop space L Ψ n ( f ) X n consists of an n -tuple of points x , . . ., x n ∈ X and paths γ i from f ( x i ) to x i + , indices modulo n . As n varies, the sus-pension spectra Σ ∞+ L Ψ n ( f ) X n form a naive restriction system (Example 8.3.8.3). Follow-ing [MP18b] and the precedent set by [BHM93], we denote its topological restrictionhomology by TR( X ; f ). Proposition 9.27.
For any path-connected X and basepoint-preserving f : X → X , thereis an equivalence of restriction systems
THH ( r ) ( S [ Ω X ]; S [ Ω f X ]) ≃ Σ ∞+ L Ψ r ( f ) X r and therefore an equivalence on TRTR( S [ Ω X ]; S [ Ω f X ]) ≃ TR( X ; f ).To prove the proposition, note that both sides of the equivalence are homotopy invari-ant, and so we may without loss of generality assume that X = BG for a cofibrant topo-logical group G , and use G as the model for Ω X . We may also assume that f : BG → BG arises by applying the classifying space functor to a group homomorphism, which byabuse of notation we also denote f : G → G .The first step is to identify(9.28) THH ( r ) ( S [ G ]; S [ G f ]) = THH( S [ G × r ]; S [ G × rf × r ρ ]) ∼= THH( S [ G × r ]; S [ G × r Ψ r ( f ) ]) by applying ρ − once to the bimodule coordinate. This gives an isomorphism of restric-tion systems where the one on the right arises from the isomorphisms ( Ψ rs ( f )) C r ∼= Ψ s ( f ).The proposition then follows from the next lemma by passage to suspension spectra. Lemma 9.29.
There is a natural equivalence of restriction systems of spacesB cy ( G × r ; G × r Ψ r ( f ) ) ≃ −→ L Ψ r ( f ) BG × r . Proof.
We start by constructing two equivalences of spaces(9.30) B cy ( G ; G f ) ≃ −→ EG × G G ad f ≃ −→ L f BG where G ad f denotes G with the left G -action g · a = f ( g ) a g − . The first equivalence isactually an isomorphism, and is defined on k -simplices by( g , . . ., g k ; g ) [ g | · · · | g k ] g g · · · g k .To define the second, we compare two different fibrant models for the base change 1-cell BG id f = [ BG f −→ BG ] in the bicategory of parametrized spaces over varying base spaces.The first is the fibrant approximation of the parametrized space (id, f ) : BG −→ BG given by the space of paths ( ev , f ◦ ev ) : BG I −→ BG . The second is EG × G G f , where G f is given the left G -action ( g , h ) a = f ( g ) ah − , and the map to BG arises from theprojection ( p , p ) : EG → BG . The equivalence(9.31) BG −→ EG × G G f , [ g | · · · | g k ] [( g , f ( g )) | · · · | ( g k , f ( g k ))] e of spaces over BG is the fibrant approximation map. We form a commuting square ofspaces over BG where the horizontal maps are induced by (9.31) and the vertical mapsare inclusion of constant paths: BG ∼ (cid:15) (cid:15) ∼ / / EG × G G f ∼ (cid:15) (cid:15) BG I ∼ / / ( EG × G G f ) I The final space projects to BG by ( p ◦ ev , p ◦ ev ). If we remove the upper left instanceof BG , the other three spaces are fibrant over BG , hence we can pull them back alongthe diagonal ∆ : BG → BG to get a zig-zag of equivalences of spaces over BG L f BG ∼ −→ ∆ ∗ h ( EG × G G f ) I i ∼ ←− EG × G G ad f .The second equivalence in (9.30) is this zig-zag.Applying the construction (9.30) to the group G × r and the map Ψ r ( f ) : G × r −→ G × r ,we have a composite equivalence B cy ( G × r ; G × r Ψ r ( f ) ) ≃ −→ EG × r × G × r G ad × r Ψ r ( f ) ≃ −→ L Ψ r ( f ) BG × r ,which defines each level of the equivalence of restriction systems in the statement of thelemma. Since both of the maps are C r -equivariant, and taking fixed points with respectto a subgroup of C r gives the same maps for a smaller value of r , the desired equivalenceof restriction systems follows. (cid:3) Remark . The map (9.30) is the canonical equivalence between two different mod-els for r ! ∆ ∗ ( BG id f ) that are computed by deriving the base change functor ∆ ∗ in twodifferent ways. It follows that each level of the equivalence of restriction systems inProposition 9.27 is a point set model for the comparison map of shadows induced bythe equivalence of symmetric monoidal bifibrations from [Mal19, §8.2] (see also [MP18a,§14]). -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 51 Recall from [MP18b] that the Fuller trace R ( Ψ n ( f )) C n is defined as the C n -equivariantReidemeister trace of the map Ψ n ( f ). By the main theorem of [MP18b], these assembleto define a class R ( Ψ ∞ ( f )) ∈ π TR( X ; f )called the infinite Fuller trace. Theorem 9.33.
The compositeK ( S [ Ω X ], S [ Ω f X ]) trc −−→ π TR( S [ Ω X ]; S [ Ω f X ]) ∼= π TR( X ; f ) of the TR -trace and the equivalence from Proposition 9.27 takes the class of the twistedmodule endomorphism [ f ] from (9.18) to the infinite Fuller trace R ( Ψ ∞ ( f )) .Proof. We examine the image of the class [ f ] under the TR-trace and the routes in thecommutative diagram π TR( S [ Ω X ]; S [ Ω f X ]) π TR( X ; f ) π THH( S [ Ω X ]; S [ Ω Ψ n ( f ) X ]) π Σ ∞+ L Ψ n ( f ) X × n π THH( S [ Ω X ]; S [ Ω f ◦ n X ]), ∼= g n g n ∼=∼= where the upper square commutes by the naturality of the ghost maps for the equiv-alence of restriction systems from Proposition 9.27, and the lower-left vertical map isthe composition of (9.28) and (9.20). Theorem 9.19 states that the image in the lowerleft corner is the Reidemeister trace R ( f ◦ n ) of the iterate. By the unwinding argumentof [MP18b, Thm. 1.1], it follows that the image in the middle-left is the Fuller trace R ( Ψ n ( f )). Therefore the image in the middle-right is R ( Ψ n ( f )) as computed in parame-trized spectra. By Proposition 9.17 the upper vertical maps are jointly injective, hencethe image of [ f ] in π TR( X ; f ) is the infinite Fuller trace R ( Ψ ∞ ( f )). (cid:3) Since the infinite Fuller trace capture the behavior of n -periodic points for every n ≥ K -theory might capture deeper dynamical information. Weplan to continue this investigation in future work.A PPENDIX
A. M
ODEL CATEGORIES OF RESTRICTION SYSTEMS
Recall that in §8 we introduced the notion of a “restriction system” to link the equi-variant Dennis traces together and define the TR-trace. A restriction system is like acyclotomic spectrum, but more general. In this appendix, we show how to move theserestriction systems between different models of spectra, and we establish a model struc-ture for pre-restriction systems in the spirit of [BM15].Recall the notion of an equivariant bispectrum from [CLM + , Appendix A]. We definegenuine restriction systems of these bispectra by using the geometric fixed point functorfrom [CLM + , Definition A.9]. Proposition A.1.
If X • is a genuine (pre-)restriction system of bispectra and each X n iscofibrant as a C n -bispectrum then the prolongation P X • to orthogonal spectra is naturallya genuine (pre-)restriction system of orthogonal spectra.Proof. We define the structure maps by Φ C r P X rs P Φ C r X rs ∼= o o P c r / / P X s . By [CLM + , Proposition A.7], P preserves equivalences of cofibrant spectra, so if the maps c r are stable equivalences then so are these new maps. To check these form a pre-restriction system it suffices to show the diagram below commutes. The top-right regioncommutes because the bispectra X • form a restriction system. The bottom-right regioncommutes by naturality and the left-hand region commutes by [CLM + , Lemma A.13]. Φ C rs P X rst it (cid:15) (cid:15) P Φ C rs X rst ∼= o o it (cid:15) (cid:15) P c rs / / P X t P Φ C s Φ C r X rst ∼= (cid:15) (cid:15) P Φ Cs c r / / P Φ C s X st P c s O O ∼= (cid:15) (cid:15) Φ C s Φ C r P X rst Φ C s P Φ C r X rst ∼= o o Φ Cs P c r / / Φ C s P X st (cid:3) Example A.2.
For every termwise cofibrant naive restriction system of symmetric spec-tra X • , the orthogonal suspension spectra Σ ∞ X • form a genuine restriction system ofbispectra. In fact, the restriction map from the categorical fixed points κ : ¡ Σ ∞ X rs ¢ C r −→ Φ C r Σ ∞ X rs is an isomorphism, so we define the maps of the genuine restriction system using theinverse of the naive restriction system maps: Φ C r Σ ∞ X rs ∼= ¡ Σ ∞ X rs ¢ C r ∼= Σ ∞ ( X C r rs ) ∼= ←− Σ ∞ X s .Their compatibility follows immediately from rigidity. Example A.3.
Similarly, for every genuine restriction system of orthogonal spectra X • ,the symmetric suspension spectra Σ ∞ X • form a genuine restriction system of bispectrawith structure maps Φ C r Σ ∞ X rs ∼= Σ ∞ Φ C r X rs Σ ∞ c r −→ Σ ∞ X s .Next we place model structures on the various categories of restriction systems. It issimple enough to do this for naive restriction systems { X n } because they are equivalentto spectra with a Z -action and no free Z -orbits. The weak equivalences and fibrationsare measured on each term X n separately, and the cofibrations are generated by themaps of naive restriction systems that are ∗ at all levels n not divisible by a , and theshift-desuspensions of the canonical inclusions F m ( S k − × Z / a Z ) + −→ F m ( D k × Z / a Z ) + at all levels n where a | n . Proposition A.4.
This defines a model structure on naive restriction systems of symmet-ric spectra.
Next we turn to the model structure on genuine restriction systems. The idea, as in[BM15, §5], is to build a model structure on genuine pre-restriction systems by thinkingof them as algebras over a monad C . As in that paper, this is not completely correctbecause C is only a monad on cofibrant inputs, but we can still use it to create freepre-restriction systems, which is enough to build the model structure.Consider the category of sequences { X n } n ≥ in which the n th term X n is a C n -equivariantorthogonal spectrum or bispectrum. Such a sequence is termwise cofibrant if each X n is cofibrant in the stable model structure. We define C { X • } to be the sequence with n thterm C { X • } n = _ m ≥ Φ C m X mn . -THEORY OF ENDOMORPHISMS, THE TR -TRACE, AND ZETA FUNCTIONS 53 If the sequence { X • } is termwise cofibrant then we make C { X • } into a genuine pre-restriction system by defining c r to be the composite Φ C r C { X • } rs C { X • } s Φ C r µ _ m ≥ Φ C m X mrs ¶ _ m ≥ Φ C r Φ C m X mrs / / ∼= _ m ≥ Φ C mr X mrs . / / ∼= O O Note that the final map is an inclusion of some but not all of the summands of C { X • } s .Hence this is a not a restriction system, only a pre-restriction system. The compatibilitycheck for the structure maps c r can be done on each summand of the source separately,where it follows from rigidity. Lemma A.5.
On cofibrant inputs, C is the left adjoint of the forgetful functor from pre-restriction systems to sequences of equivariant spectra.Proof. A map of pre-restriction systems C X −→ Y is given by maps of C n -spectra f m , n : Φ C m X mn −→ Y n for all m , n ≥ c r . In the case of r = m , onefinds that the compatibility condition implies that the maps f n : X n → Y n determine allof the others. (cid:3) The generating cofibrations for the model structure on pre-restriction systems areconstructed using the sets I n of generating cofibrations for C n -spectra, for all n ≥
1, byconsidering each map as a morphism of equivariant sequences that is only nontrivial atthe n th term, and then applying C to get a map of pre-restriction systems. Concretely,these are the maps of pre-restriction systems that are trivial on the k th term unless k | n , in which case they are given by the maps of C k -spectra Φ C n / k F ( m , V ) ³ C n / C a × S k − ´ + −→ Φ C n / k F ( m , V ) ³ C n / C a × D k ´ + .Call the collection of such maps C I . By the preservation properties of geometric fixedpoints detailed in [CLM + , Lemma A.10], any C I -cell complex is at the n th term of therestriction system an I n -cell complex.We perform the same construction to the generating acyclic cofibrations, and in thecase of bispectra to the generating cofibrations for the model structure on C n -equivariantbispectra from [CLM + , Proposition A.5]. Verifying that our definitions define a modelstructure is now straightforward by checking the properties termwise. Proposition A.6.
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