A geometric approach to equivariant factorization homology and nonabelian Poincaré duality
aa r X i v : . [ m a t h . A T ] A ug A GEOMETRIC APPROACH TO EQUIVARIANT FACTORIZATIONHOMOLOGY AND NONABELIAN POINCAR ´E DUALITY
FOLING ZOU
Abstract.
In this paper, we use the minimal categorical background and maximalconcreteness to study equivariant factorization homology in the V -framed case. Wework with a finite group G and an n -dimensional orthogonal G -representation V . Weset up a monadic bar construction for the equivariant factorization homology for a V -framed manifold with coefficients in E V -algebra. We then prove the nonabelianPoincar´e duality theorem using a geometrically-seen scanning map. Contents
1. Introduction 21.1. Factorization homology: history and equivariant 21.2. Nonabelian Poincar´e duality theorem 41.3. Notations 41.4. Acknowledgement 52. Preliminary 52.1. Λ-sequences and operads 52.2. Equivariant bundles 83. Tangential structures and factorization homology 133.1. Equivariant tangential structures 133.2. Equivariant factorization homology 163.3. Relation to configuration spaces 194. Nonabelian Poincar´e Duality for V -framed manifolds 234.1. Scanning map for V -framed manifolds 234.2. Nonabelian Poincar´e duality theorem 244.3. Connectedness 264.4. Cofibrancy 274.5. Dimension 29Appendix A. A comparison of scanning maps 33A.1. Scanning map from tubular neighborhood 33A.2. Scanning map using geodesic 35A.3. Scanning equivalence 37Appendix B. A comparison of θ -framed morphisms 38B.1. The θ -framed maps 38B.2. Automorphism space of ( V, φ ) 42References 44 Introduction
Factorization homology is a homology theory on manifolds with coefficients in suitableE n -algebras. The main results of this paper are:(1) We construct a G Top-enriched category Mfld fr V n . Its objects are V -framed G -manifolds of dimension n . The endomorphism operad of the object V is equiva-lent to the little V -disk operad of Guillou-May [GM17], thus it is an E V -operad.(2) With this category, we define the equivariant factorization homology Z M A bya monadic bar construction.(3) We prove the nonabelian Poincar´e duality theorem using a geometrically-seenscanning map, which establishes a weak G -equivalence Z M A ≃ Map ∗ ( M + , B V A ).Here, M is a V -framed manifold, and M + is its one-point compactification. The coef-ficient A is an E V -algebra and B V A is the V -fold deloop of A .The approach in this paper follows the non-equivariant treatment in [Mil15]. Itis a global generalization of the delooping machines of [May72, GM17]. The non-abelian Poincar´e duality theorem gives a simplicial filtration on the mapping spaceMap ∗ ( M + , B V A ), thus offering a calculational tool.There are other approaches of a different flavor to equivariant factorization homol-ogy, developed by [Hor19, Wee18]. In joint work with Horev and Klang, we give analternative proof of the nonabelian Poincar´e duality theorem in [HHK +
20] in Horev’scontext, together with an application to Thom G -spectra.1.1. Factorization homology: history and equivariant.
The language of factor-ization homology has been used to formulate and solve questions in many areas ofmathematics. Among others, there are homological stability results in [KM18, Knu18],a reconstruction of the cyclotomic trace in [AMGR17] and the study of quantum fieldtheory in [BZBJ18, CG16].Non-equivariantly, factorization homology has multiple origins. The most well-knownapproach started in Bellinson-Drinfeld’s study of an algebraic geometry approach toconformal field theory [BD04] under the name of Chiral Homology. Lurie [Lur, 5.5]and Ayala–Francis [AF15] introduced and extensively studied the algebraic topologyanalogue, named as factorization homology. This route relies heavily on ∞ -categoricalfoundations. An alternative geometric model is Salvatore’s configuration spaces withsummable labels [Sal01]. This construction is close to the geometric intuition, butnot homotopical. Yet another model, using the bar construction and developed byAndrade [And10], Miller [Mil15] and Kupers–Miller [KM18], is homotopically well-behaved while staying close to the geometric intuition of configuration spaces.We take the third approach in this paper. To give an idea of the concept, we startwith the non-equivariant story.Classically, the Dold–Thom theorem states that the symmetric product is a homol-ogy theory. For a based CW-complex M with base point ∗ , the symmetric producton M is Symm( M ) = (cid:0) ` k ≥ M k / Σ k (cid:1) / ∼ , where ∼ is the base-point identification( m , · · · , m k , ∗ ) ∼ ( m , · · · , m k ). The Dold-Thom theorem states that when M is con-nected, there are natural isomorphisms π ∗ (Symm( M )) ∼ = e H ∗ ( M, Z ) . QUIVARIANT FACTORIZATION HOMOLOGY 3
Factorization homology, viewed as a functor on manifolds, generalizes the homologytheory of topological spaces. It uses the manifold structure to work with coefficients inthe noncommutative setting. Essentially, Z M A is the configuration space on M withsummable labels in an E n -algebra A ; the local Euclidean chart offers the way to sumthe labels. Rigorously, [And10, KM18] defines the factorization homology on M to be:(1.1) Z M A = B(D M , D n , A ) , where D n is the reduced monad associated to the little n -disks operad and D M is thefunctor associated to embeddings of disks in M .This bar construction definition is a concrete point-set level model of the ∞ -categoricaldefinition [Lur, AF15]. We can construct a topological category Mfld fr n of framed smooth n -dimensional manifolds and framed embeddings. It is a symmetric monoidal categoryunder taking disjoint union. Let Disk fr n be the full subcategory spanned by objectsequivalent to ⊔ k R n for some k ≥
0. An E n -algebra A is just a symmetric monoidaltopological functor out of Disk fr . The factorization homology is the derived symmetricmonoidal topological left Kan extension of A along the inclusion:(1.2) Disk fr n (Top , × )Mfld fr n A R − A Horel [Hor13, 7.7] shows the equivalence of (1.1) and (1.2).We could also view factorization homology as a functor on the algebra. This gives ageometric interpretation of some classical invariants of structured rings and a way toproduce more. For example, THH of an associative ring is equivalent to the factorizationhomology on S . We will not make use of this perspective in this paper.We point out one technicality of this bar construction that takes some effort equiv-ariantly, namely, how to give the morphism space of Mfld fr n . On the one hand, from thedefinition of the little V -disks operad, we want to include only the “rectilinear” embed-dings as framed embeddings; on the other hand, we lose control of any rectilinearilyonce we throw the little disks into the wild category of framed manifolds.The solution is to allow all embeddings but add in path data to correct the homotopytype. This idea goes back to Steiner [Ste79] where he used paths to construct anespecially useful E n -operad. Andrade [And10] and Kupers–Miller [KM18] used pathsin the framing space to define framed embedding spaces so that they do not see theunwanted rotations.An equivariant version of E n -algebra is given by Guillou–May’s little V -disks operad D V and E V -algebras in [GM17]. The E V -algebras give the correct coefficient input ofequivariant factorization homology on V -framed manifolds.In Section 3.1, we construct the category Mfld fr V n of V -framed smooth G -manifoldsof dimension n . A V -framing of M is a trivialization φ M : T M ∼ = M × V of itstangent bundle. We put the V -framing into a general framework of tangential structures θ : B → B G O ( n ) and define the θ -framed embedding space of θ -framed manifolds usingpaths in the framing space. In Appendix B, we compare this approach to an alternativeone that does not make explicit use of the framing space. FOLING ZOU
In Section 3.2, we use the G Top-enriched category Mfld fr V n to build V -framed factor-ization homology by a monadic bar construction. The ingredients to set up the barconstruction are the V -framed little disks operad D fr V V , the monad D fr V V and the functorD fr V M , which is a right module over D fr V V (Definition 3.11). Definition 1.3. (Definition 3.13) The equivariant factorization homology is: Z M A = B (D fr V M , D fr V V , A ) . In Section 3.3, we study the homotopy type of the defined embedding space in Mfld fr V n and show that Emb fr V ( ` k V, M ), the V -framed embedding space, has the same homo-topy type as F M ( k ), the ordered configuration space of k points in M : Theorem 1.4. (Corollary 3.28(1)) Evaluating at of the embedding gives a ( G × Σ k ) -homotopy equivalence: ev : Emb fr V ( a k V, M ) ≃ → F M ( k ) . In particular, the V -framed little disks operad is equivalent to the Guillou–May little V -disks operad (Proposition 3.31), so it is an E V -operad.1.2. Nonabelian Poincar´e duality theorem.
Our main theorem is:
Theorem 1.5. (Theorem 4.8) Let M be a V -framed manifold and A be a G -connected D fr V V -algebra in G Top . There is a weak G -equivalence: Z M A ≃ Map ∗ ( M + , B V A ) . The proof of Theorem 1.5 is inspired by [Mil15]. There are two main ingredients: therecognition principal in [May72, GM17] for the local result, and the scanning map thathas been studied non-equivariantly in [McD75, BM88, MT14].In Section 4.1, we construct the scanning map, a natural transformation of rightD fr V V -functors: s : D fr V M ( − ) → Map c ( M, Σ V − ) , and compare it to the scanning maps in the literature in Appendix A.In Section 4.2 to Section 4.5, we prove Theorem 1.5.1.3. Notations. • G Top is the Top-enriched category of G -spaces and G -equivariant maps. • Top G is the G Top-enriched category of G -spaces and non-equivariant mapswhere G acts by conjugation on the mapping space. QUIVARIANT FACTORIZATION HOMOLOGY 5
We note the following facts:(1) G Top is the underlying Top-enriched category of Top G : G Top(
X, Y ) ∼ = G Top(pt , Top G ( X, Y )) . (2) G Top is a closed Cartesian monoidal category. The interal hom G -space is givenby the morphism in Top G .For orthogonal G -representations V and W , we use the following notations for themapping spaces, all of which are G -subspaces of Top G ( V, W ): • Hom(
V, W ) for linear maps; • Iso(
V, W ) for linear isomorphisms of vector spaces; • O( V, W ) for linear isometries; • O( V ) for O( V, V ).For a space X and b ∈ X , • P b X is the path space of X at the base point b ; • Ω b X is the loop space of X at the base point b ; • Λ b X is the Moore loop space of X at the base point b , defined to be Λ b X = { ( l, α ) ∈ R ≥ × X R ≥ | α (0) = b, α ( t ) = b for t ≥ l } . For a space X , a vector space V and a map φ : V → X , • Ω φ X is Ω φ (0) X ; Λ φ X is Λ φ (0) X .For based spaces X, Y and an unbased space M , • Map ∗ ( Y, X ) is the space of based maps; • Map c ( M, X ) = { f ∈ Map(
M, X ) | f − ( X \ ∗ ) is compact } is the space of com-pactly supported maps.For a space M and a fiber bundle E → M , • F M ( k ) is the ordered configuration space of k points in M . • F E ↓ M ( k ) is the ordered configuration space of k points in E whose images are k distinct points in M .1.4. Acknowledgement.
This paper is the main part of my thesis. I would like toexpress my deepest gratitude to my advisor Peter May, who raises me up from thekindergarten of mathematics. I am indebted to Inbar Klang, whose work motivates mythesis, and to Alexander Kupers and Jeremy Miller, whose work leads to the approachin my research. I would like to thank Haynes Miller for helpful conversations andShmuel Weinberger for being my committee member.2.
Preliminary -sequences and operads.
To streamline the monadic bar construction in themain body, we study unital operads, reduced monads and bar constructions using anelementary categorical framework of Λ-objects in more details in a separate paper withMay and Zhang [MZZ20]. This subsection is a summary of the relevant content andreaders familiar with operads may skip it.Let Λ be the category of based finite sets n = { , , , · · · , n } with base point 0 andbased injections. The morphisms of Λ are generated by permutations and the orderedinjections s ki : k − → k that skip i for 1 ≤ i ≤ k . It is a symmetric monoidal category FOLING ZOU with wedge sum as the symmetric monoidal product. Let ( V , ⊗ , I ) be a bicompletesymmetric monoidal category with initial object ∅ , terminal object ∗ . Let V I be thecategory under the unit. Later we will mostly be concerned about ( G Top , × , pt) whichis Cartesian monoidal, so G Top pt = G Top ∗ is the category of pointed G -spaces. Definition 2.1.
A Λ-sequence in V is a functor E : Λ op → V . It is called unital if E ( ) = I . The category of all Λ-sequences in V is denoted Λ op ( V ), where morphismsare natural transformations of functors. The category of all unital Λ-sequences in V is denoted Λ op I ( V ), where morphisms are natural transformations of functors that areidentity at level zero.The category Λ op [ V ] admits a symmetric monoidal structure (Λ op [ V ] , ⊠ , I ). It isthe Day convolution of functors on the closed symmetric monoidal category Λ op . Theunit is given by I ( n ) = ( I , n = 0; ∅ , n > op [ V ] I for the category of objects under the unit I . The symmetricmonoidal product ⊠ on Λ op [ V ] induces a symmetric monoidal product on Λ op [ V ] I andits subcategory Λ op I [ V ], which we still denote by ⊠ . Remark 2.2.
To clarify a possible confusion with notation, note that E ∈ Λ op I [ V ] isa unital Λ-sequence with E ( ) = I , while F ∈ Λ op [ V ] I comes with a specified map I → F ( ). F is called a unitary Λ-sequence in [MZZ20].Both categories highlighted in the remark above admit a (nonsymmetric) monoidalproduct ⊙ in addition to ⊠ . It is analogous to Kelly’s circle product on symmetricsequences [Kel05]. The unit for ⊙ is given by I ( n ) = ( I , n = 0 , ∅ , n > I (1) → I (0) is the identity. For a brief defini-tion of ⊙ , see Construction 2.9 (2); for a detailed definition, see [MZZ20].For a Λ-sequence E , the spaces E ( k ) admit Σ k -actions, so E has an underlyingsymmetric sequence. Though not relevant to this paper, it is surprising that the Dayconvolution of Λ-sequences agrees with the Day convolution of symmetric sequences: Theorem 2.3. ( [MZZ20, Theorem 3.3] ) For D , E ∈ Λ op [ V ] , there is an isomorphismof symmetric sequences D ⊠ Σ E → D ⊠ Λ E . Of course, Kelly’s circle product on symmetric sequences does not agree with the circleproduct on Λ-sequences.An operad in V , as defined in [May97], gives an example of a symmetric sequencein V . If the operad is unital, meaning the 0-space of the operad is the unit, it hasthe structure of a Λ-sequence in V . In fact, We have the unital variant of Kelly’sobservation [Kel05]: Theorem 2.4. ( [MZZ20, Theorem 0.10] ) A unital operad in V is a monoid in themonoidal category (Λ op I [ V ] , ⊙ , I ) . QUIVARIANT FACTORIZATION HOMOLOGY 7
When V = Top or V = G Top, a unital operad is also called a reduced operad in[May97].We give a construction which will be used in the definition of equivariant factorizationhomology: the associated functor of a unital Λ-sequence. This construction specializesto the reduced monad associated to a reduced operad of [May97] when V is Cartesianmonoidal; it also appears in the definition of the circle product ⊙ . Construction 2.5.
Let ( W , ⊗ , J ) be a symmetric monoidal category and X ∈ W J bean object under the unit. Define X ∗ : Λ → W to be the covariant functor that sends n to X ⊗ n . On morphisms, it sends the permutations to permutations of the X ’s andsends the injection s ki : k − → k for 1 ≤ i ≤ k to the map( s ki ) ∗ : X ⊗ k − ∼ = X ⊗ i − ⊗ J ⊗ X ⊗ k − i X ⊗ k , id i − ⊗ η ⊗ id k − i where η : J → X is the unit map of X . By convention, X ⊗ = J .This defines a functor ( − ) ∗ : W J → Fun ⊗ (Λ , W ). Here, Fun ⊗ (Λ , W ) is the categoryof strong symmetric monoidal functors from Λ to W . Remark 2.6.
The above defined functor ( − ) ∗ is indeed an equivalence with an inversegiven by the forgetful functor Fun ⊗ (Λ , W ) → W J that sends X to X ( ).Assume that ( W , ⊗ , J ) is a cocomplete symmetric monoidal category tensored over V . Then one can form the categorical tensor product over Λ of the contravariant functor E and the covariant functor X ∗ . Construction 2.7.
Let E ∈ Λ op [ V ] I be a unitary Λ-sequence. The functorE : W J → W J associated to E is defined to beE( X ) = E ⊗ Λ X ∗ = a k ≥ E ( k ) ⊗ X ⊗ k / ≈ , where ( α ∗ f, x ) ≈ ( f, α ∗ x ) for all f ∈ E ( m ), x ∈ X ⊗ n and α ∈ Λ( n , m ). The unit mapof E( X ) is given by J ∼ = I ⊗ J → E (0) ⊗ X ⊗ → E( X ). Remark 2.8.
It is sometimes useful to take the quotient in two steps and use thefollowing alternative formula for E:E( X ) = a k ≥ E ( k ) ⊗ Σ k X ⊗ k / ∼ , where [( s ki ) ∗ f, x ] ∼ [ f, ( s ki ) ∗ x ] for all f ∈ E ( k ), x ∈ X ⊗ k − . We will use ≈ or ∼ for theequivalence relation to be clear which formula we are using and refer to ∼ as the basepoint identification. Construction 2.9.
We focus on the following context of Construction 2.7.(1) Let W = V . The associated functor is E : V I → V I . In particular, taking V = G Top, one gets for a reduced G -operad C ∈ Λ op ∗ ( G Top) the reduced monad
C : G Top ∗ → G Top ∗ . FOLING ZOU (2) Let W = (Λ op [ V ] , ⊠ , I ) via the Day monoidal structure. Then W is tensoredover V in the obvious way by levelwise tensoring. One gets the circle product for E ∈ Λ op [ V ] I and F ∈ Λ op [ V ] I by: E ⊙ F := E ⊗ Λ F ∗ ∈ Λ op [ V ] I . These two cases are further related: the 0-th level functor ı : V → Λ op [ V ] , ( ı X )( n ) = ( X, n = 0; ∅ , n > ı (E X ) ∼ = ı ( E ⊗ Λ X ∗ ) ∼ = E ⊗ Λ ( ı ( X ) ∗ ) ∼ = E ⊙ ı X. In words, the reduced monad construction is what happens at the 0-space of the circleproduct. Using this, one can show
Proposition 2.11. ( [MZZ20, Proposition 6.2] ) Let E , F : V I → V I be the functorsassociated to E and F . Then the functor associated to E ⊙ F is E ◦ F . A monad is a monoid in the functor category. Using the associativity of the circleproduct and (2.10), it is easy to prove that when C is an operad, the associated functorC in Construction 2.7 is a monad.The following construction gives examples of monoids and modules in (Λ op I [ V ] , ⊙ ): Construction 2.12. ([MZZ20, Section 8]) Suppose that we have a V -enriched sym-metric monoidal category ( W , ⊗ , I W ) such that W ( I W , Y ) ∼ = I V for all objects Y of W . Then we can construct a Λ op I V [ V ]-enriched category H W . The objects are the sameas those of W , while the enrichment is given by H W ( X, Y ) = W ( X ⊗∗ , Y ) . The definition of the composition in H W is similar to the structure maps of an endomor-phism operad. So, for any objects X, Y, Z of W , H W ( Y, Y ) is monoid in (Λ op I [ V ] , ⊙ ), H W ( X, Y ) is a left module over it, and H W ( Y, Z ) is a right module. In the light ofTheorem 2.4, H W ( Y, Y ) is a unital operad, the endomorphism operad. The assumption W ( I W , Y ) ∼ = I V is automatically satisfied if W is coCartesian monoidal.We will use that the circle product is strong symmetric monoidal in the first variable: Proposition 2.13. ( [MZZ20, Proposition 4.7] ) For any E ∈ Λ op [ V ] I , the functor − ⊙ E on (Λ op ( V ) I , ⊠ , I ) is strong symmetric monoidal. That is, the circle productdistributes over the Day convolution: for any D , D ′ ∈ Λ op ( V ) I , we have ( D ⊠ D ′ ) ⊙ E ∼ = ( D ⊙ E ) ⊠ ( D ′ ⊙ E ) . Equivariant bundles.
In this paper, we characterize the V -framing of a G -manifold and the space of V -framed maps using equivariant bundles. This approachhas the advantage of being very concrete. In this subsection, we list some preliminaryresults for the reader’s reference, with a focus on vector bundles. The proofs as wellas a clarification of different notions of equivariant fiber bundles can be found in thecompanion paper [Zou20b].Let G and Π be compact Lie groups, where G is the ambient action group and Π isthe structure group. QUIVARIANT FACTORIZATION HOMOLOGY 9
Definition 2.14. A G - n -vector bundle a map p : E → B such that the followingstatements hold:(1) The map p is a non-equivariant n -dimensional vector bundle;(2) Both E and B are G -spaces and p is G -equivariant;(3) The G -action is linear on fibers. Definition 2.15.
A principal G -Π-bundle is a map p : P → B such that the followingstatements hold:(1) The map p is a non-equivariant principal Π-bundle;(2) Both P and B are G -spaces and p is G -equivariant;(3) The actions of G and Π commute on P . Remark 2.16.
This is called a principal ( G, Π)-bundle in [LMSM86, IV1].As in the non-equivariant case, we write the Π-action on the right of a principal G -Π-bundle P ; for convenience of diagonal action, we consider P to have a left Π-action,that is, ν ∈ Π acts on z ∈ P by νz = zν − . Theorem 2.17.
There is an equivalence of categories between { G - n -vector bundles over B } and { principal G - O ( n ) -bundles over B } . To deal with more general cases than G -vector bundles, for example, Atiyah’s Realvector bundles, tom Dieck [TD69] introduced a complex conjugation action of C onthe structure group U ( n ). Lashof–May [LM86] had the idea to further introduce a totalgroup that is the extension of the structure group Π by G . Tom Dieck’s work becamea special case of a split extension, or equivalently a semidirect product. Definition 2.18. ([LM86]) Let 1 → Π → Γ → G → p : P → B such that the followingstatements hold:(1) The map p is a non-equivariant principal Π-bundle;(2) The space P is a Γ-space; B is a G -space. Viewing B as a Γ-space by pullingback the action, the map p is Γ-equivariant.A morphism between two principal (Π; Γ)-bundles p : P → B and p : P → B is apair of maps ( ¯ f , f ) fitting in the commutative diagram P P B B fp p f such that f is G -equivariant and ¯ f is Γ-equivariant.Taking the extension to be Γ = Π × G , we recover the principal G -Π-bundles ofDefinition 2.15. In this case we have two names for the same thing. This could beconfusing, but since a “principal G -Π-bundle” looks more natural than a “principal(Π; Π × G )-bundle” for this thing, we will keep both names.There is also a structure theorem identifying the category of equivariant principalbundles of Definition 2.18 with suitable equivariant fiber bundles: Theorem 2.19. ( [LMSM86, IV1] ) For any Π -effective Γ -space F and G -space B , thereis an equivalence of categories between { G -fiber bundles with structure group Π , totalgroup Γ and fiber F over B } and { principal (Π; Γ) -bundles over B } . There are two subtleties here: One is that the fiber F needs to have a preassignedΓ-action; the other is how to define the structure group of a fiber bundle. We skipthe details and the interested reader may refer to the original reference or [Zou20b] forexplanations.2.2.1. V -trivial bundles. A G -vector bundle E → B is V -trivial for some n -dimensional G -representation V if there is a G -vector bundle isomorphism E ∼ = B × V . Such an iso-morphism is a V -framing of the bundle. This is analogous to the case of non-equivariantvector bundles, except that equivariance adds in the complexity of a representation V that’s part of the data.However, the representation V in the equivariant trivialization of a fixed vector bundlemay not be unique. Let Iso( V, W ) be the space of linear isomorphisms V → W withthe conjugation G -action for G -representations V and W . Lemma 2.20.
For a G -space B , there exists a G -vector bundle isomorphism B × V ∼ = B × W if and only if there exists a G -map f : B → Iso(
V, W ) . Corollary 2.21. If B has a G -fixed point, then B × V ∼ = B × W only when V ∼ = W . Example 2.22 (Counterexample) . Let G = C , σ be the sign representation. The unitsphere, S (2 σ ), is S with the 180 degree rotation action. As C -vector bundles, S (2 σ ) × R ∼ = S (2 σ ) × σ. Example 2.23. (Counterexample, Gus Longerman) Take G to be any compact Liegroup and V and W to be any two representation of G that are of the same dimension.Then G × V ∼ = G × W .2.2.2. V -framing bundles. Equivariantly, G -representations serve the role of vector spacesand there can be more than one of them in each dimension. So it is natural to considerthe V -framing bundle for an orthogonal n -dimensonal representation V . Definition 2.24.
Let p : E → B be a G - n -vector bundle. Let Fr V ( E ) be the space ofthe admissible maps with the G -action g ( ψ ) = gψρ ( g ) − . In other words, Fr V ( E ) has the same underlying space as Fr R n ( E ), but we think ofadmissible maps as mapping out of V instead of R n . One would expect Fr V ( E ) issome principal bundle in the sense of Definition 2.18. Although this is true, it does notcapture the complete data.Let V be given by ρ : G → O ( n ). We write O ( V ) for the group O ( n ) with the data G → Aut( O ( n )) given by g ( ν ) = ρ ( g ) νρ ( g ) − for g ∈ G and ν ∈ O ( n ), so it is clearwhat O ( V ) ⋊ G means. This convention coincides with the conjugation G -action on O ( V ) thought of as a mapping space. Proposition 2.25. Fr V ( E ) is a principal ( O ( V ); O ( V ) ⋊ G ) -bundle and we have iso-morphisms of G -vector bundles: E ∼ = (Fr V ( E ) × V ) /O ( n ) . QUIVARIANT FACTORIZATION HOMOLOGY 11
Note that we have an isomorphism(2.26) O ( V ) ⋊ G ∼ = O ( n ) × G ( ν, g ) ↔ ( νρ ( g ) , g ) . So Fr V ( E ) and Fr R n ( E ) are the same even as principal (Π; Γ)-bundles, where(Π; Γ) = ( O ( V ); O ( V ) ⋊ G ) ∼ = ( O ( n ) , O ( n ) × G ) . The G -actions tell them apart in two perspectives. When 1 → Π → Γ → G → G → Γ,which gives a G -action on the total space of a principal (Π; Γ)-bundle. These are the G -actions on Fr R n ( E ) and Fr V ( E ). The isomorphism (2.26) is not compatible with thesplitting, resulting in the different G -actions. In fact, G in the second line of (2.27)includes as the graph subgroup Λ ρ = { ( ρ ( g ) , g ) | g ∈ G } ⊂ O ( n ) × G .(2.27) 1 O ( n ) O ( n ) × G G O ( V ) O ( V ) ⋊ G G ∼ =(2 .
26) ( e,g ) ← [ g ( e,g ) ← [ g The second perspective to see the difference of Fr V ( E ) and Fr R n ( E ) is via the dif-ferent G -actions on the fiber R n to recover E in Proposition 2.25. Recall that thefiber of an equivariant fiber bundle should have an action of the extended structuregroup Γ (See Theorem 2.19); for split extensions this is equivalent to specifying a G -action. To recover E from Fr V ( E ), the fiber with the appropriate G -action is exactlythe representation V thought of as a G -space.2.2.3. Fixed points.
Let H ⊂ G be a subgroup and Rep( H, Π) be the set:Rep( H, Π) = { group homomorphism ρ : H → Π } / Π-conjugation . A group homomorphism ρ : H → Π gives a subgroup Λ ρ ⊂ (Π × G ) via its graph:Λ ρ = { ( ρ ( h ) , h ) | h ∈ H } . Denote the centralizer of the image of ρ in Π by Z Π ( ρ ). We haveZ Π ( ρ ) = Π ∩ Z Π × G (Λ ρ ) = { ν ∈ Π | νρ ( h ) = ρ ( h ) ν for all h ∈ H } . [LM86, Theorem 12] gives complete information on the fixed-point spaces of a prin-cipal (Π; Γ)-bundle. We focus on the special case of the trivial extension Γ = Π × G when a principal (Π; Γ)-bundle is just a principal G -Π-bundle. Take p : P → B to besuch a principal G -Π-bundle. Then Lashof–May’s quoted theorem associates to eachcomponent B ⊂ B H a homomorphism [ ρ ] ∈ Rep( H, Π):
Theorem 2.28. { ρ : H → Π | (cid:0) p − ( B ) (cid:1) Λ ρ = ∅ } form a single conjugacy class ofrepresentations. Furthermore, the (non-equivariant) principal Π -bundle p − ( B ) → B has a reduction of the structure group from Π to the closed subgroup Z Π ( ρ ) ⊂ Π . Note that a bundle morphism preserves the associated representation [ ρ ]. Also, E Λ ρ → p ( E Λ ρ ) is a principal Z Π ( ρ )-bundle for a fixed representation ρ . Equivariant classifying spaces.
The universal principal (Π; Γ)-bundle can be rec-ognized by the following property:
Theorem 2.29. ( [LM86, Theorem 9] ) A principal (Π; Γ) -bundle p : E → B is universalif and only if E Λ ≃ ∗ , for all subgroups Λ ⊂ Γ such that Λ ∩ Π = e. When Γ = Π × G , such a subgroup Λ comes in the form of Λ ρ as defined above. Notation 2.30.
The universal principal G - O ( n )-bundle is denoted E G O ( n ) → B G O ( n );The universal principal (Π; Γ)-bundle is denoted E (Π; Γ) → B (Π; Γ). We denote theuniversal G - n -vector bundle by ζ n → B G O ( n ), where ζ n = E G O ( n ) × O ( n ) R n . From information about the fixed-point spaces and Theorem 2.29, one gets the G -homotopy type of the universal base: Theorem 2.31. ( [Las82, Theorem 2.17] ) ( B G O ( n )) G ≃ a [ ρ ] ∈ Rep(
G,O ( n )) B Z O ( n ) ( ρ ); ≃ a [ V ] ∈ Rep(
G,O ( n )) B ( O ( V ) G ) . Example 2.32.
Take H = G = C and Π = O (2). ThenRep( C , O (2)) = { id , rotation , reflection } . For ρ = id or ρ = rotation, Z Π ( ρ ) = O (2). For ρ = reflection, Z Π ( ρ ) ∼ = Z / × Z /
2. So( B C O ( n )) C ≃ BO (2) ⊔ BO (2) ⊔ B ( Z / × Z / . One can make explicit the classifying maps of V -trivial bundles as follows. A G -map θ : pt → B G O ( n ) gives the following data: it lands in one of the G -fixed components of B G O ( n ), indexed by a representation class [ V ]; its image is a G -fixed point b ∈ B G O ( n ). Proposition 2.33.
The pullback of the universal bundle is θ ∗ ζ n ∼ = V . The loop space of B G O ( n ) at the base point b , Ω b B G O ( n ), is a G -space with thepointwise G -action on the loops. Via concatenation of loops, it is an A ∞ -algebra in G -spaces. Using the Moore loop space Λ b B G O ( n ), whose definition we omit here, wecan strictify Ω b B G O ( n ) to a G -monoid. Definition 2.34. A G -monoid is a monoid in G -spaces, that is, an underlying monoidsuch that the multiplication is G -equivariant. A morphism of G -monoids is an equiva-lence if it is a weak G -equivalence. Theorem 2.35.
Let O ( V ) be isometric self maps of V with G acting by conjugation.(1) There is a G -homotopy equivalence Ω b B G O ( n ) ≃ O ( V ) ;(2) There is an equivalence of G -monoids Λ b B G O ( n ) ≃ O ( V ) . Theorem 2.35 is used later in Theorem B.14 for understanding the automorphismspace of a framed disk V . QUIVARIANT FACTORIZATION HOMOLOGY 13
Remark 2.36.
Explicitly, the equivalence of G -monoids is given by a zigzag(2.37) Λ b B G O ( n ) ( e Λ b E G O ( n )) /O ( n ) O ( V ) . ξ ≃ ψ ≃ Here, let p denote the universal principal G - O ( n )-bundle E G O ( n ) → B G O ( n ); We define e Λ b E G O ( n ) = { ( l, α ) | l ∈ R ≥ , α : R ≥ → E G O ( n ) , p ( α (0)) = p ( α ( t )) = b for t ≥ l } , so that ( e Λ b E G O ( n )) /O ( n ) = [ l, α ] where the equivalence relation is( l, α ) ∼ ( l, β ) if there is ν ∈ O ( n ) such that α ( t ) = β ( t ) ν for all t ≥ . While e Λ b E G O ( n ) does not have the structure of a G -monoid, ( e Λ b E G O ( n )) /O ( n ) does.Fix a base point z ∈ p − ( b ) ⊂ E G O ( n ). The maps are given by ξ ([ l, α ]) = ( l, p ( α )) ∈ Λ b B G O ( n ); ψ ([ l, α ]) ∈ O ( V ) is determined by α (0) ψ ([ l, α ]) = α ( l ) . We conclude this section with results on the morphism spaces of equivariant principalbundles. Let 1 → Π → Γ → G → p Π;Γ : E (Π; Γ) → B (Π; Γ)be the universal principal (Π; Γ)-bundle and let p : P → B be any principal (Π; Γ)-bundle. Let Hom( P, E (Π; Γ)) be the space of (non-equivariant) principal Π-bundlemorphisms. Since Hom(
P, E (Π; Γ)) ∼ = Map Π ( P, E (Π; Γ)), the conjugation Γ-action onMap(
P, E (Π; Γ)) descends to give a G -action on Hom( P, E (Π; Γ)). One can prove:
Lemma 2.38.
Hom(
P, E (Π; Γ)) is G -contractible. Lemma 2.38 leads to the following result. Let p : P → B be any principal G - O ( n )-bundle. Restricting a bundle map to its map of base spaces gives(2.39) π : Hom( P, E G O ( n )) → Map p ( B, B G O ( n )) . Here, Map p ( B, B G O ( n )) is the (non-equivariant) component of the classifying map of p in Map( B, B G O ( n )); G acts by conjugation on both sides of (2.39). Note that G actson Aut B P by conjugation, so we can form Γ = Aut B P ⋊ G . Theorem 2.40.
The map (2.39) is a universal principal (Aut B P ; Γ) -bundle. Tangential structures and factorization homology
Equivariant tangential structures.
Fix an integer n and a finite group G . Atangential structure is a G -map θ : B → B G O ( n ). Here, B G O ( n ) is the classifyingspace as in Notation 2.30. A morphism of two tangential structures is a G -map over B G O ( n ). All tangential structures form a category T S , which is simply the over cate-gory G Top / B G O ( n ) .In this subsection we fix a tangential structure θ and construct two categories. Thefirst one is Vec θ , the category of n -dimensional θ -framed bundles with θ -framed bundlemaps as morphisms. The second one is Mfld θ , the category of smooth n -dimensional θ -framed manifolds and θ -framed embeddings. The category Mfld θ is a subcategory ofVec θ ; both Mfld θ and Vec θ are enriched over G Top. If we let θ vary, both constructionsdefine covariant functors from T S to categories.
Denote by ζ n the universal G - n -vector bundle over B G O ( n ). Pulling back along thetangential structure θ : B → B G O ( n ) gives a bundle θ ∗ ζ n over B . This is meant tobe the universal θ -framed vector bundle. For an n -dimensional smooth G -manifold M ,the tangent bundle of M is a G - n -vector bundle. It is classified by a G -map up to G -homotopy: τ : M → B G O ( n ) . Definition 3.1. A θ -framing on a G - n -vector bundle E → M is a G - n -vector bundlemap φ E : E → θ ∗ ζ n . A θ -framing on a smooth G -manifold M is a θ -framing φ M on itstangent bundle. We abuse notations and refer to the map on the base spaces as φ M aswell.A bundle has a θ -framing if and only if its classifying map τ : M → B G O ( n ) has afactorization up to G -homotopy through B ; see diagram (3.2). However, a factorization τ B : M → B does not uniquely determine a θ -framing φ E : E → θ ∗ ( ζ n ). Indeed, abundle map φ E : E → θ ∗ ( ζ n ) is the same data as a map τ B : M → B on the base plus ahomotopy between the two classifying maps from M to B G O ( n ). For a detailed proof,see Corollary B.9 with Definition B.4.(3.2) BM B G O ( n ) θττ B h y Example 3.3.
When B is a point, a tangential structure θ : pt → B G O ( n ) picks outin its image a G -fixed component of B G O ( n ). This component is indexed by some n -dimensional G -representation V up to isomorphism. Then θ ∗ ζ n ∼ = V as a G -vector spaceover pt (Proposition 2.33). We denote this tangential structure by fr V : pt → B G O ( n )and call it a V -framing. A V -framing on a vector bundle E → M is just an equivarianttrivialization E ∼ = M × V . We emphasize that the V -framing tangential structure isnot only a space B = pt but also a map fr V . Definition 3.4.
Given two θ -framed bundles E , E with framings φ , φ , the space of θ -framed bundle maps between them is defined as:(3.5) Hom θ ( E , E ) := hofib (cid:0) Hom( E , E ) φ ◦− −→ Hom( E , θ ∗ ζ n ) (cid:1) , where Hom( E , θ ∗ ζ n ) is based at φ . Explicitly, a θ -framed bundle map is a bundle map f and a homotopy connecting the two resulting θ -framings φ and φ f of E :Hom θ ( E , E ) = { ( f, α ) ∈ Hom( E , E ) × Hom( E , θ ∗ ζ n ) I | α (0) = φ , α (1) = φ f } . The unit in Hom θ ( E, E ) is given by (id E , φ const ); the composition of two θ -bundle mapsis defined as: Hom θ ( E , E ) × Hom θ ( E , E ) → Hom θ ( E , E ); (cid:0) ( g, β ) , ( f, α ) (cid:1) ( g ◦ f, λ ) , where λ ( t ) = (cid:26) α (2 t ) , when 0 ≤ t ≤ / β (2 t − ◦ f, when 1 / < t ≤ . QUIVARIANT FACTORIZATION HOMOLOGY 15
As defined above, the composition is unital and associative only up to homotopy.One can modify Hom θ ( E , E ) using Moore paths in the homotopy to make the com-position strictly unital and associative; see [KM18, Definition 17] or Definition B.4 fora construction in the same spirit. We omit the details here and assume we have builta category Vec θ of θ -framed bundles and θ -framed embeddings. As such, an elementof Hom θ ( E , E ) has a third data of the length of the path, which is a locally constantfunction on Hom( E , E ), but for brevity we sometimes do not write it.In the definition of Hom θ ( E , E ), everything is taken non-equivariantly. The spacesHom( E , E ) and Hom( E , θ ∗ ζ n ) have G -actions by conjugation. Since φ and φ are G -maps, the homotopy fiber Hom θ ( E , E ) inherits the conjugation G -action. Definition 3.6.
The space of θ -framed embeddings between two θ -framed manifoldsis defined as the pullback displayed in the following diagram of G -spaces:(3.7) Emb θ ( M, N ) Hom θ (T M, T N )Emb( M, N ) Hom(T M, T N ) d Here, Emb(
M, N ) is the space of smooth embeddings and the map d takes an embeddingto its derivative. Remark 3.8.
Most of the time, we drop the Moore-path-length data and write anelement of Emb θ ( M, N ) as a package of a map f and a homotopy ¯ f = ( f, α ), with f ∈ Emb(
M, N ) and α : [0 , → Hom(T M, T N ) satisfying α (0) = φ M and α (1) = φ N ◦ df . There is a functor Mfld θ → Mfld by forgetting the tangential structure. Itsends ¯ f ∈ Emb θ ( M, N ) to f ∈ Emb(
M, N ).Let ⊔ be the disjoint union of θ -framed vector bundles or manifolds and ∅ be theempty bundle or manifold. Both (Vect θ , ⊔ , ∅ ) and (Mfld θ , ⊔ , ∅ ) are G Top-enrichedsymmetric monoidal categories. In both categories, ∅ is the initial object. In Vect θ , ⊔ is the coproduct, but not in Mfld θ . Remark 3.9.
We need the length of the Moore path to be locally constant as introducedin [KM18, Definition 17] as opposed to constant for the enrichment to work. Namely,the map Hom θ ( E , E ′ ) × Hom θ ( E , E ′ ) → Hom θ ( E ⊔ E , E ′ ⊔ E ′ )is given by first post-composing with the obvious θ -framed map E ′ i → E ′ ⊔ E ′ for i = 1 ,
2, then using a homeomorphism, as follows:Hom θ ( E , E ′ ) × Hom θ ( E , E ′ ) → Hom θ ( E , E ′ ⊔ E ′ ) × Hom θ ( E , E ′ ⊔ E ′ ) ∼ = Hom θ ( E ⊔ E , E ′ ⊔ E ′ )If the length of the Moore path were constant, the displayed homeomorphism wouldonly be a homotopy equivalence, as the length of a Moore path can be different on thetwo parts.To set up factorization homology in Section 3.2, we fix an n -dimensional orthogonal G -representation V ; in addition, we suppose that V is θ -framed and fix a θ -framing φ : T V → θ ∗ ζ n on V . Since T V ∼ = V as G -vector bundles, the space of θ -framings on V is(3.10) Hom(T V, θ ∗ ζ n ) G ≃ Hom(
V, θ ∗ ζ n ) G = Hom( R n , θ ∗ ζ n ) Λ ρ ≃ ( θ ∗ E G O ( n )) Λ ρ , where Λ ρ = { ( ρ ( g ) , g ) ∈ O ( n ) × G | g ∈ G } and ρ : G → O ( n ) is a matrix representationof V . By Theorem 2.28, ( θ ∗ E G O ( n )) Λ ρ ∼ = θ ∗ ( E G O ( n )) Λ ρ . So the spaces in (3.10) are non-empty, or a θ -framing on V exists, if and only if theintersection of θ ( B ) and the V -indexed component of ( B G O ( n )) G as introduced inTheorem 2.31 is non-empty.We also describe the change of tangential structures, which is not studied in thispaper. Let q be a morphism from θ : B → B G O ( n ) to θ : B → B G O ( n ), equivalently,a G -map q : B → B satisfying θ q = θ . Then a θ -framed vector bundle E → B with φ E : E → θ ∗ ζ n is θ -framed by E → θ ∗ ζ n = q ∗ θ ∗ ζ n → θ ∗ ζ n . The morphism q also induces a map on framed-morphisms. So we have a functor q ∗ : Vec θ → Vec θ , and similarly q ∗ : Mfld θ → Mfld θ . Equivariant factorization homology.
In this subsection, we use the Λ-sequencemachinery in Section 2.1 and the G Top-enriched category Mfld θ developed in Section 3.1to define the equivariant factorization homology as a bar construction.Recall from Section 3.1 that we have fixed an n -dimensional orthogonal G -representation V and a θ -framing φ : T V → θ ∗ ζ n on the G -manifold V . Definition 3.11.
For a θ -framed manifold M , we define the Λ-sequence D θM to be D θM = Emb θ ( ∗ V, M ) ∈ Λ op ∗ ( G Top) . Here, ∗ V is the symmetric monoidal functor (Λ , ∨ , ) → (Mfld θ , ⊔ , ∅ ) that sends to( V, φ ) and sends → to the unique map ∅ ֒ → V .Explicitly, on objects, we have D θM : Λ op → G Top , k Emb θ ( a k V, M );On morphisms, Σ k = Λ( k , k ) acts by permuting the copies of V , and s ki : k − → k induces ( s ki ) ∗ : D θM ( k ) → D θM ( k −
1) by forgeting the i -th V in the embeddings for1 ≤ i ≤ k .Plugging in V in the second variable, we have D θV . Using Construction 2.9, we getassociated functors of D θM and D θV , which we denote byD θM , D θV : G Top ∗ → G Top ∗ ;D θM ( X ) = a k ≥ D θM ( k ) × Σ k X × k / ∼ . These Λ-sequences satisfy certain structures coming from the composition of mor-phisms in Mfld θ . It is best described using the Kelly monoidal structure (Λ op ∗ ( G Top) , ⊙ ) QUIVARIANT FACTORIZATION HOMOLOGY 17 as defined in Construction 2.9. Taking V = G Top and ( W , ⊗ ) = (Mfld θ , ⊔ ) in Construction 2.12,we can identify D θM = H W ( V, M ) . Consequently, D θV = H W ( V, V ) is a monoid in (Λ op ∗ ( G Top) , ⊙ ) and D θM is a right moduleover it.Translating by Theorem 2.4, D θV is a reduced operad in ( G Top , × ). This operad isclose to the little V -disk operad D V except it also allows θ -framed automorphisms ofthe embedded V -disks. We make the remark that in light of Theorem B.14, we expectthere to be something like an equivalence of G -operads: D θV ≃ D V ⋊ ( Λ φ B ). This isformulated and proved in [Zou20a, Appendix B].By Proposition 2.11, the right module map D θM ⊙ D θV → D θM of Λ-sequences yieldsa natural transformation D θM ◦ D θV → D θM ; The monoid structure maps I → D θV and D θV ⊙ D θV → D θV yield natural transformations id → D θV and D θV ◦ D θV → D θV .The following is a standard definition from [May97]: Definition 3.12.
Let C be a reduced operad in ( G Top , × ) and C be the associatedreduced monad. An object A ∈ G Top ∗ is a C -algebra if there is a map γ : C A → A such that the following diagrams commute, where the unlabeled maps are the unit andmultiplication map of the monad C: CCA CACA A
Cγ γγ ; A CAA γ . In what follows, let A be a D θV -algebra in G Top ∗ . We have a simplicial G -space,whose q -th level is B q (D θM , D θV , A ) = D θM (D θV ) q A. The face maps are induced by the above-given structure mapsD θM D θV → D θM , D θV D θV → D θV and γ : D θV A → A. The degeneracy maps are induced by id → D θV .We have the following definition after the idea of [And10, IX.1.5]: Definition 3.13.
The factorization homology of M with coefficient A is Z θM A := B (D θM , D θV , A ) . Notation 3.14.
Since we are not comparing tangential structures in this paper, wedrop the θ in the notation and write Z θM A as Z M A .The category of algebras D θV [ G Top ∗ ] has a transfer model structure via the forgetfulfunctor D θV [ G Top ∗ ] → G Top ∗ ([BM03, 3.2, 4.1]), so that weak equivalences of mapsbetween algebras are just underlying weak equivalences. Proposition 3.15.
The functor Z M − : D θV [ G Top ∗ ] → G Top ∗ is homotopical. Proof.
The proof is a formal argument assembling the literature and deferred. Weshow that the bar construction is Reedy cofibrant in Corollary 4.22. The claim thenfollows since geometric realization preserves levelwise weak equivalences between Reedycofibrant simplicial G -spaces, as quoted in Theorem 4.15. (cid:3) We have the following properties of the factorization homology.
Proposition 3.16. Z V A ≃ A. Proof.
This follows from the extra degeneracy argument of [May72, Proposition 9.8].The extra degeneracy coming from the unit map of the first D θV establishes A as aretract of B (D θV , D θV , A ), which is just Z V A. (cid:3) Proposition 3.17.
For θ -framed manifolds M and N , Z M ⊔ N A ≃ Z M A × Z N A. Proof.
Without loss of generality, we may assume that both M and N are connected.Then D θM ⊔ N ( k ) ∼ = Emb θ ( ⊔ k V, M ⊔ N ) ∼ = k a i =0 (cid:0) Emb θ ( ⊔ i V, M ) × Emb θ ( ⊔ k − i V, N ) (cid:1) × Σ i × Σ k − i Σ k ∼ = k a i =0 (cid:0) D θM ( i ) × D θN ( k − i ) (cid:1) × Σ i × Σ k − i Σ k This is the formula of the Day convolution of D θM and D θN . So we have(3.18) D θM ⊔ N ∼ = D θM ⊠ D θN . We drop the θ in the rest of the proof. By (3.18) and iterated use of Proposition 2.13,there is an isomorphism in Λ op ∗ ( G Top) for each q :(3.19) B q ( D M ⊔ N , D V , ı ( A )) ∼ = B q ( D M , D V , ı ( A )) ⊠ B q ( D N , D V , ı ( A )) . Iterated use of (2.10) identifies ı ( B q (D M , D V , A )) ∼ = B q ( D M , D V , ı ( A )) , so evaluating on the 0-th level of (3.19) gives equivalence of simplical G -spaces: B ∗ (D M ⊔ N , D V , A ) ∼ = B ∗ (D M , D V , A ) × B ∗ (D N , D V , A ) . The claim follows from passing to geometric realization and commuting the geometricrealization with finite products. (cid:3)
QUIVARIANT FACTORIZATION HOMOLOGY 19
Relation to configuration spaces.
Now we restrict our attention to the V -framed case for an orthogonal n -dimensional G -representation V . We give V the canon-ical V -framing T V ∼ = V × V and let M be a G -manifold of dimension n . When M is V -framed, we denote the V -framing by φ M : T M → V .In this subsection, we first prove that a smooth embedding of ⊔ k V into M is deter-mined by its images and derivatives at the origin up to a contractible choice of homotopy(Proposition 3.24). The proof of the non-equivariant version can be found in Andrade’sthesis [And10, V4.5]. Then we proceed to prove that a V -framed embedding space of ⊔ k V into M as defined in (3.7) is homotopically the same as choosing the center points(Corollary 3.28).To formulate the result, we first define the suitable equivariant configuration spacerelated to a manifold, which will be “the space of points and derivatives”.We use F E ( k ) to denote the ordered configuration space of k distinct points in E ,topologized as a subspace of E k . When E is a G -space, F E ( k ) has a G -action bypointwise acting that commutes with the Σ k -action by permuting the points. Definition 3.20.
For a fiber bunde p : E → M , define F E ↓ M ( k ) to be configurationsof k -ordered distinct points in E with distinct images in M . F E ↓ M ( k ) is a subspaceof F E ( k ) and inherits a free Σ k -action. When p is a G -fiber bundle, F E ↓ M ( k ) is a G -space. Example 3.21.
When k = 1, F E ↓ M (1) ∼ = F E (1). Example 3.22.
When E = M × F is a trivial bundle over M with fiber F , F E ↓ M ( k ) ∼ = F M ( k ) × F k . In general, we have the following pullback diagram: F E ↓ M ( k ) E k F M ( k ) M k . p k Now, we take E = Fr V (T M ). Recall that Fr V (T M ) = Hom( V, T M ) is a G -bundleover M . For an embedding ⊔ k V → M , we take its derivative and evaluate at 0 ∈ V .We will get k -points in Fr V (T M ) with different images projecting to M . In other words,the compositionEmb( a k V, M ) d → Hom( a k T V, T M ) ev → Hom( a k V, T M ) = Fr V (T M ) k factors as(3.23) Emb( a k V, M ) d → F Fr V (T M ) ↓ M ( k ) ֒ → Fr V (T M ) k . Proposition 3.24.
The map d in (3.23) is a G -Hurewicz fibration and ( G × Σ k ) -homotopy equivalence.Proof. It suffices to prove for k = 1, that is, for d : Emb( V, M ) → Fr V (T M ) , since the general case will follow from the pullback diagram:Emb( ` k V, M ) Emb(
V, M ) k F Fr V (T M ) ↓ M ( k ) Fr V (T M ) k . d ( d ) k We show that d is a G -Hurewicz fibration by finding an equivariant local trivial-ization. Fix an H -fixed point x ∈ Fr V (T M ) and let d − ( x ) be the fiber at x . Ourgoal is to find an H -invariant neighborhood ¯ U of x in Fr V (T M ) and an H -equivarianthomeomorphism ¯ U × d − ( x ) ∼ = d − ( ¯ U ) ⊂ Emb(
V, M ) . First, we find the small neighborhood ¯ U . Let x be the image of x under the pro-jection Fr V (T M ) → M , then x is also H -fixed. Consequently, W = T x M is an H -representation. Using the exponential map, there is a local chart for M that is H -homeomorphic to W with 0 ∈ W mapping to x . We will refer to this local chart as W .On the chart, Fr V (T M ) is homeomorphic to W × Hom(
V, W ), and we may identify x with (0 , A ) ∈ W × Hom(
V, W ) for some H -invariant A . For simplicity, we put a metricon W to make it an orthogonal representation. Choose an ǫ -ball U ⊂ W and a smallenough H -invariant neighborhood A ∈ U ⊂ Hom(
V, W ) and set ¯ U = U × U .Second, we construct an H -equivariant local trivialization of d on ¯ U ,¯ φ : ¯ U × d − ( x ) → Emb(
V, M ) , ( y, f ) φ ( y ) ◦ f by utilizing a yet-to-be-constructed map φ : ¯ U → Diff( M ). The map φ needs to satisfythe following properties:(1) φ is H -equivariant;(2) φ ( x ) = id;(3) For any y ∈ ¯ U , d ( φ ( y )) ◦ x = y . (Recall that x, y ∈ Fr V (T M ) = Hom( V, T M )and d ( φ ( y )) : T M → T M is the derivative of φ ( y ).)For any χ ∈ Diff( M ) and g ∈ Emb(
V, M ), d ( χ ◦ g ) = d g (0) ( χ ) ◦ d ( g ). We can check that d ( φ ( y ) ◦ f ) = y and that for any g ∈ Emb(
V, M ) with d ( g ) = y , d ( φ ( y ) − ◦ g ) = x .So, the map φ ( y ) ◦ − translates d − ( x ), the fiber over x , to d − ( y ), the fiber over y .This shows ¯ φ is an H -equivariant homeomorphism to d − ( ¯ U ).Third, we describe only the idea of the construction of φ , as it is a bit technical.Noticing that the requirement (3) is local, we can construct φ : ¯ U → Diff( W ) onthe local chart W satisfying all the requirements using linear maps. Then we need tomodify these diffeomorphisms of W equivariantly without changing them on the ǫ -ball U , so that they become compactly supported and still satisfy all the requirements.Finally, we extend the modified φ by identity to get φ , diffeomorphisms of M . Thetechnical part is the modification for φ . It can be done by (1) taking an H -invariantpolytope P containing U , (2) taking a large enough multiple m such that mP containsthe image of all φ ( ¯ U )( U ), (3) setting φ ( y ) to be id outside of mP , (4) extending bypiecewise linear function between P and mP , and (5) smoothing it. It is because of thisstep that we have to choose a small enough neighborhood U , but it is good enough forour purpose. QUIVARIANT FACTORIZATION HOMOLOGY 21
To show d is a G -homotopy equivalence, one can construct a section of d by theexponential map: σ : Fr V (T M ) → Emb(
V, M ) . Since there is a (contractible) choice of the radius at each point for the exponential mapto be homeomorphism, σ is defined only up to homotopy. Using blowing-up-at-origintechniques, the section can be shown to indeed give a deformation retract of d .To be useful later, the section exists up to homotopy for general k as well: (cid:3) (3.25) σ : F Fr V (T M ) ↓ M ( k ) → Emb( a k V, M ) . Now we are ready to justify our desired equivalence of the V -framed embedding spacesfrom V to M and configuration spaces of M . Moreover, we show that this equivalenceis compatible over Emb( ` k V, M ) in part (2). This will be used in later sections tocompare different scanning maps.
Lemma 3.26.
For a V -framed manifold M , the projection F Fr V (T M ) ↓ M ( k ) → F M ( k ) is a trivial bundle with fiber (Hom( V, V )) k . We call the section that selects (id V ) k ineach fiber the zero section z .Proof. Regarding V as a bundle over a point, we may identify Fr V ( V ) = Hom( V, V ).Since M is V -framed, Fr V (T M ) ∼ = Fr V ( M × V ) ∼ = M × Fr V ( V ) as equivariant bundles.The claim follows from Example 3.22. (cid:3) We can restrict the exponential map (3.25) to the zero section in Lemma 3.26 to get(3.27) σ : F M ( k ) → Emb( a k V, M ) . Corollary 3.28.
For a V -framed manifold M , we have:(1) Evaluating at of the embedding gives a ( G × Σ k ) -homotopy equivalence: ev : D fr V M ( k ) ≡ Emb fr V ( a k V, M ) → F M ( k ) . (2) The map ev and σ in (3.27) fit in the following ( G × Σ k ) -homotopy commutativediagram: Emb fr V ( ` k V, M ) Emb( ` k V, M ) F M ( k ) ev σ Proof. (1) By Definitions 3.6 and 3.11, Emb fr V ( ` k V, M ) is the homotopy fiber of thecomposite: D : Emb( a k V, M ) d → Hom( a k T V, T M ) ( φ M ) ∗ → Hom( a k T V, V ) . We would like to restrict the composite at { } ⊔ · · · ⊔ { } ⊂ V ⊔ · · · ⊔ V . SinceHom( a k T V, T M ) ∼ = Y k Hom(T V, T M ) and i : V → T V is a G -homotopy equivalence of G -vector bundles, ev : Hom( a k T V, T M ) ( i ) ∗ → Y k Hom( V, T M ) ∼ = (Fr V (T M )) k is a ( G × Σ k )-homotopy equivalence. So in the following commutative diagram, thevertical maps are all ( G × Σ k )-homotopy equivalences:Emb( ` k V, M ) Hom( ` k T V, T M ) Hom( ` k T V, V ) F Fr V (T M ) ↓ M ( k ) Fr V (T M ) k Fr V ( V ) k F M ( k ) × Fr V ( V ) k Fr V ( V ) k . dd ≃ by Proposition 3.24 ev ≃ ( φ M ) ∗ ev ≃∼ = by Lemma 3.26 ( φ M ) ∗ proj We focus on the top composition D and the bottom map proj . The map ev betweentheir codomains is a based map. Indeed, the base point of Hom( ` k T V, V ) is from the V -framing of ` k V and is ( G × Σ k )-fixed. It is mapped to id k , the base point of Fr V ( V ) k .Consequently, there is a ( G × Σ k )-homotopy equivalence between the homotopy fibersof these two maps.(3.29) Emb fr V ( a k V, M ) = hofib( D ) ≃ → hofib( proj ) . Our desired ev in question is the composite of (3.29) and the following map: X : hofib( proj ) → F M ( k ) × Fr V ( V ) k proj → F M ( k ) . It suffices to show that X is a ( G × Σ k )-equivalence. Indeed, X is the comparison ofthe homotopy fiber and the actual fiber of proj . Write temporarily F = F M ( k ) and B = Fr V ( V ) k with the ( G × Σ k )-fixed base point b . Then the map X is projection to F : hofib( proj ) ∼ = P b B × F → F. The claim follows from the fact that P b B is ( G × Σ k )-contractible.(2) We have the following ( G × Σ k )-homotopy commutative solid diagram, where z is the zero section in Lemma 3.26:Emb fr V ( ` k V, M ) Emb( ` k V, M ) F M ( k ) F Fr V (T M ) ↓ M ( k ) . ev d zσ The commutativity can be seen easily and is actually an extension of the big commu-tativity diagram in part (1) to (homotopy) fibers. As σ = σ ◦ z and σ is a ( G × Σ k )-homotopy inverse of d by Proposition 3.24, the diagram with the dotted arrow ishomotopy commutative. (cid:3) Remark 3.30.
Part (1) of Corollary 3.28 gives a levelwise equivalence of objects inΛ op ∗ ( G Top): ev : D fr V M → F M . QUIVARIANT FACTORIZATION HOMOLOGY 23
We conclude this subsection by comparing D fr V V to D V . For background, the little V -disks operad D V is a well-studied notion introduced for recognizing V -fold loop spaces;see [GM17, 1.1]. It is an equivariant generalization of the little n -disks operad. Roughlyspeaking, D V ( k ) is the space of non-equivariant embeddings of k copies of the open unitdisks D( V ) to D( V ), each of which takes only the form v a v + b for some 0 < a ≤ b ∈ D( V ), called rectilinear. In particular, the spaces are the same as those oflittle n -disks operad, and so are the structure maps. The G -action on D V ( k ) is byconjugation. It is well-defined, commutes with the Σ k -action and the structure mapsare G -equivariant. Proposition 3.31.
There is an equivalence of G -operads β : D V → D fr V V .Proof. To construct the map of operads β , we first define β (1) : D V (1) → D fr V V (1). Take e ∈ D V (1), we must give β (1)( e ) = ( f, l, α ) ∈ D fr V V (1). Explicitly, e : D( V ) → D( V ) is e ( v ) = a v + b for some 0 < a ≤ b ∈ D( V ) . Define f : V → V to be f ( v ) = a v + b ; l ∈ R ≥ to be l = − ln( a ); α : R ≥ → Hom(T
V, V ) to be α ( t ) = ( c exp( − t )I for t ≤ l ; c a I for t > l. For α , Hom(T V, V ) ∼ = Map( V, O ( V )), I is the unit element of O ( V ) and c is the constantmap to the indicated element. It can be checked that β (1) as defined is a map of G -monoids.Restricting β (1) k : D V (1) k → D fr V V (1) k to the subspace D V ( k ) ⊂ D V (1) k , we get β ( k ) : D V ( k ) → D fr V V ( k ). Then β is automatically a map of G -operads because D V and D fr V V are suboperads of D V (1) − and ( D fr V V ) − .The composite ev ◦ β : D V → D fr V V → F V is a levelwise homotopy equivalence by[GM17, Lemma 1.2]. We have shown ev is a levelwise equivalence (Remark 3.30). So β is also a levelwise homotopy equivalence. (cid:3) Nonabelian Poincar´e Duality for V -framed manifolds Configuration spaces have scanning maps out of them. It turns out that equivariantlythe scanning map is an equivalence on G -connected labels X . Since the factorizationhomology is built up simplicially by the configuration spaces, we can upgrade the scan-ning equivalence to what is known as the nonabelian Poincar´e duality theorem.4.1. Scanning map for V -framed manifolds. In this subsection we construct thescanning map, a natural transformation of right D fr V V -functors:(4.1) s : D fr V M ( − ) → Map c ( M, Σ V − ) . In Appendix A, we compare our scanning map to the existing different constructionsin the literature and utilize known results about equivariant scanning maps to giveTheorem 4.5, a key input to the nonabelian Poincar´e duality theorem in Section 4.2.Assuming that the scanning map (4.1) has been constructed for a moment, the rightD fr V V -functor structure for Map c ( M, Σ V − ) is as follows: the scanning map for M = V gives a map of monads s : D fr V V → Ω V Σ V . It induces a natural mapΣ V D fr V V Σ V s −→ Σ V Ω V Σ V counit −→ Σ V . Now we construct the scanning map. For any G -space X , recall thatD fr V M ( X ) = a k ≥ D fr V M ( k ) × Σ k X k / ∼ , where ∼ is the base point identification. Take an element P = [ ¯ f , · · · , ¯ f k , x , · · · , x k ] ∈ D fr V M ( k ) × Σ k X k . Here, each ¯ f i = ( f i , α i ) consists of an embedding f i : V → M and a homotopy α i of twobundle maps T V → V , see Definition 3.6. We use only the embeddings f i to define anelement s X ( P ) ∈ Map c ( M, Σ V X ):(4.2) s X ( P )( m ) = ( f − i ( m ) ∧ x i when m ∈ M is in the image of some f i ; ∗ otherwise.Notice that if x i is the base point, f − i ( m ) ∧ x i is the base point regardless of what f i is. So passing to the quotient, (4.2) yields a well-defined map(4.3) s X : D fr V M ( X ) → Map c ( M, Σ V X ) . In particular, taking X = S , we get(4.4) s S : a k ≥ D fr V M ( k ) / Σ k → Map c ( M, S V ) , and s X is simply a labeled version of it. A more categorical construction of the scanningmap s X , as the composition of the Pontryagin-Thom collapse map and a “folding” map ∨ k S V × X k → Σ V X is given in [MZZ20, Section 9].We use the following results of Rourke–Sanderson [RS00], which are proved usingequivariant transversality. To translate from their context to ours, see Theorem A.2and Theorem A.12. Theorem 4.5.
The scanning map s X : D fr V M X → Map c ( M, Σ V X ) is:(1) a weak G -equivalence if X is G -connected,(2) or a weak group completion if V ∼ = W ⊕ and M ∼ = N × R . Here, W is a ( n − -dimension G -representation and N is a W -framed compact manifold, sothat N × R is V -framed. Nonabelian Poincar´e duality theorem.
We have seen that the scanning mapis an equivalence for G -connected labels X . Since the factorization homology is builtup simplicially by the configuration spaces, we can upgrade the scanning equivalenceto what is known as the nonabelian Poincar´e duality theorem (NPD). The proof in thissubsection follows the non-equivariant treatment by Miller [Mil15].Let A be a D fr V V -algebra in G Top throughout this subsection. Assume that A isnon-degenerately based, meaning that the structure map D fr V V (0) = pt → A gives anon-degenrate base point of A . This is a mild assumption for homotopical purposes.We use the following V -fold delooping model of A . QUIVARIANT FACTORIZATION HOMOLOGY 25
Definition 4.6.
The V -fold delooping of A , denoted as B V A , is the monadic two sidedbar construction B(Σ V , D fr V V , A ).Here, B q (Σ V , D fr V V , A ) = Σ V (D fr V V ) q A . The first face map Σ V D fr V V → Σ V is induced bythe scanning map of monads D fr V V → Ω V Σ V . The last face map D fr V V A → A is thestructure maps of the algebra. The middle face maps and degeneracy maps are inducedby the structure map of the monad D fr V V D fr V V → D fr V V and Id → D fr V V . Remark 4.7.
There is an equivalence of G -operads D V → D fr V V from the little V -diskoperad to the little V -framed disk operad. So a D fr V V -algebra restricts to a D V -algebraand there is an equivalence from the Guillou–May delooping [GM17] to our delooping:B(Σ V , D V , A ) → B(Σ V , D fr V V , A ) Theorem 4.8. (NPD) Let M be a V -framed manifold and A be a D fr V V -algebra in G Top .Then there is a G -map, which is a weak G -equivalence if A is G -connected: Z M A ≡ | B • (D fr V M , D fr V V , A ) | → Map ∗ ( M + , B V A ) , where M + is the one-point-compactification of M .Proof. We will sketch the proof, assuming some lemmas that are proven in the remainderof this subsection. First, from (4.1), we have a scanning map for each q ≥ fr V M (D fr V V ) q A → Map c ( M, Σ V (D fr V V ) q A ) . They assemble to a simplicial scanning map, which is a levelwise weak G -equivalenceas shown in Corollary 4.14:(4.9) B( s, id , id) : B • (D fr V M , D fr V V , A ) → Map c ( M, Σ V (D fr V V ) • A ) . One can identify the space of compactly supported maps with the space of based mapsout of the one point compactification:Map c ( M, Σ V (D fr V V ) • A ) ∼ → Map ∗ ( M + , Σ V (D fr V V ) • A ) . With some cofibrancy argument in Theorem 4.15 and Corollary 4.22, this map inducesis a weak G -equivalence on the geometric realization:B(D fr V M , D fr V V , A ) → | Map ∗ ( M + , Σ V (D fr V V ) • A ) | . Next, we change the order of the mapping space and the geometric realization. Thereis a natural map: | Map ∗ ( M + , Σ V (D fr V V ) • A ) | → Map ∗ ( M + , | Σ V (D fr V V ) • A | ) . Theorem 4.29, taking X = M + and K • = Σ V (D fr V V ) • A , gives a sufficient connectivitycondition for it to be a weak G -equivalence. This connectivity condition is then checkedin Lemma 4.35.Finally, | Σ V (D fr V V ) • A | = B V A by Definition 4.6. This finishes the proof of the theo-rem. (cid:3) Remark 4.10.
If we take M = V in the theorem and use Proposition 3.16, we get that A ≃ Ω V B V A for a G -connected E V -algebra A . This recovers [GM17, Theorem 1.14]and justifies the definition of B V A . Connectedness.Definition 4.11. A G -space X is G -connected if X H is connected for all subgroups H ⊂ G .To show that the scanning map is an equivalence in each simplicial level, we need: Lemma 4.12. If X is G -connected, then D fr V V X is also G -connected.Proof. By Corollary 3.28, D fr V V X is G -homotopy equivalent to F V X . It suffices to showthat F V X is G -connected. Fix any subgroup H ⊂ G ; we must show that ( F V X ) H isconnected. This is the space of H -equivariant unordered configuration on V with basedlabels in X . Intuitively, this is true because the space of labels X is G -connected, sothat one can always move the labels of a configuration to the base point. Nevertheless,we give a proof here by carefully writing down the fixed points of F V X in terms of thefixed points of F V ( k ) and X . We have:( F V X ) H = ( a k ≥ F V ( k ) × Σ k X k / ∼ ) H = a k ≥ ( F V ( k ) × Σ k X k ) H / ∼ H Here, ∼ is the equivalence relation in Remark 2.8 and ∼ H is ∼ restricted on H -fixedpoints. They are explicitly forgetting a point in the configuration if the correspondinglabel is the base point in X . Notice that taking H -fixed points will not commute with ≈ in Construction 2.7, but commutes with ∼ . This is because the H -action preservesthe filtration and ∼ only identifies elements of different filtrations.Since the Σ k -action is free on F V ( k ) × X k and commutes with the G -action, we havea principal G -Σ k -bundle F V ( k ) × X k → F V ( k ) × Σ k X k . To get H -fixed points on the base space, we need to consider the Λ α -fixed points onthe total space for all the subgroups Λ α ⊂ G × Σ k that are the graphs of some grouphomomorphisms α : H → Σ k . More precisely, by Theorem 2.28, we have( F V ( k ) × Σ k X k ) H = a [ α : H → Σ k ] (cid:16) ( F V ( k ) × X k ) Λ α /Z Σ k ( α ) (cid:17) . Here, the coproduct is taken over Σ k -conjugacy classes of group homomorphisms and Z Σ k ( α ) is the centralizer of the image of α in Σ k .We would like to make the expression coordinate-free for k . A homomorphism α can be identified with an H -action on the set { , · · · , k } . For an H -set S , write X S =Map( S, X ) and F V ( S ) = Emb( S, V ). Then( F V ( k ) × X k ) Λ α = ( F V ( S ) × X S ) H and Z Σ k ( α ) = Aut H ( S ) . So we have:( F V ( k ) × Σ k X k ) H = a [ S ]:iso classes of H -set , | S | = k (cid:16) ( F V ( S ) × X S ) H / Aut H ( S ) (cid:17) . If we take care of the base point identification, we end up with:(4.13) ( F V X ) H = (cid:18) a [ S ]:iso classes of finite H -set ( F V ( S ) × X S ) H / Aut H ( S ) (cid:19) / ∼ H . QUIVARIANT FACTORIZATION HOMOLOGY 27
Suppose that the H -set S breaks into orbits as S = ∐ i r i ( H/K i ) for i = 1 , · · · , s ,where K i ’s are in distinct conjugacy classes of subgroups of H and r i >
0, then weknow explicitly each coproduct component is:( F V ( S ) × X S ) H / Aut H S = (Emb H ( S, V ) × Map H ( S, X )) / Aut H S = (Emb H ( ∐ i r i ( H/K i ) , V ) × Y i ( X K i ) r i ) / Y i ( W H ( K i ) ≀ Σ r i ) . Since X K i are all connected, so are the spaces Q i ( X K i ) r i . Each contains the base pointof the labels ∗ = Q i Q r i ∗ → Q i ( X K i ) r i . So after the gluing ∼ H , each component in(4.13) is in the same component as the base point of F V X . Thus ( F V X ) H is connected. (cid:3) Corollary 4.14.
The map B • (D fr V M , D fr V V , A ) → Map c ( M, Σ V (D fr V V ) • A ) in (4.9) is alevelwise weak G -equivalence of simplicial G -spaces if A is G -connected.Proof. This is a consequence of Theorem 4.5 and Lemma 4.12. (cid:3)
For geometric realization, we have:
Theorem 4.15 (Theorem 1.10 of [MMOar]) . A levelwise weak G -equivalence betweenReedy cofibrant simplicial objects realizes to a weak G -equivalence. Cofibrancy.
We take care of the cofibrancy issues in this part, following detailsin [May72]. We first show that some functors preserve G -cofibrations. One who iswilling take it as a blackbox may skip directly to Definition 4.20. The NDR data givea hands-on way to handle cofibrations. Definition 4.16 (Definition A.1 of [May72]) . A pair (
X, A ) of G -spaces with A ⊂ X isan NDR pair if there exists a G -invariant map u : X → I = [0 ,
1] such that A = u − (0)and a homotopy given by a map h : I → Map G ( X, X ) satisfying • h ( x ) = x for all x ∈ X ; • h t ( a ) = a for all t ∈ I and a ∈ A ; • h ( x ) ∈ A for all x ∈ u − [0 , h, u ) is said to a representation of ( X, A ) as an NDR pair. A pair (
X, A ) ofbased G -spaces is an NDR pair if it is an NDR pair of G -spaces with the h t being basedmaps for all t ∈ I .Such a pair gives a G -cofibration A → X . The function u gives an open neigh-boorhood U of A by taking U = u − [0 , h restricts on I × U to aneighborhood deformation retract of A in X . We refer to u as the neighborhood dataand h as the retract data.We have the following “ ad hoc definition” for a functor F to preserve NDR-pairs ina functorial way: Definition 4.17 (Definition A.7 of [May72]) . A functor F : G Top → G Top is admissi-ble if for any representation ( h, u ) of (
X, A ) as an NDR pair, there exists a representation(
F h, F u ) of (
F X, F A ) as an NDR pair such that:(i) The map
F h : I → Map G ( F X, F X ) is determined by (
F h ) t = F ( h t ).(ii) The map F u : F X → [0 ,
1] satisfies the following property: for any map g : X → X such that ug ( x ) < x ∈ X and u ( x ) <
1, (
F u )( F g )( y ) < y ∈ F X and (
F u )( y ) < And similarly for functors F : G Top ∗ → G Top ∗ .In plain words, the retract data F h for (
F X, F A ) are dictated by applying the functor F to h , but there is some room in choosing the neighborhood data F u . Denote theopen neighborhood of
F A in F X by U ′ = ( F u ) − [0 , U ′ isa “robust open neighborhood” in the sense that a map of pairs g : ( X, U ) → ( X, U )induces a map
F g : (
F X, U ′ ) → ( F X, U ′ ). Remark 4.18.
Suppose that F sends inclusions to inclusions and that we have ( F h, F u )satisfying (i) and (ii). • In order for (
F h, F u ) to be a representation of (
F X, F A ) as an NDR pair, weonly need to check(
F u ) − (0) = F A, ( F u ) − [0 , ⊂ ( F h ) − ( F A ) . • Since we have U ⊂ h − ( A ), we get F U ⊂ F ( h − A ) ⊂ ( F h ) − ( F A ). That is,the neighborhood
F U of F A retracts to
F A , but it may not be open.Admissible functors obviously preserve cofibrations. The elaboration of the NDRdata gives a way to easily verify that a functor is admissible, at least in the followingcases:
Lemma 4.19.
Any functor F associated to F ∈ Λ op ∗ ( G Top) is admissible. In partic-ular, both D fr V V and D fr V M are admissible. The functors Map c ( M, − ) and Map ∗ ( M + , − ) are admissible. The functor Σ V sends NDR pairs to NDR pairs.Proof. To show F is admissible, it suffices to find the neighborhood data F u in eachcase.Let F ∈ Λ op ∗ ( G Top) be a unital Λ-sequence. The functor F associated to F asdefined in Construction 2.7 sends X ∈ G Top ∗ to F X = (cid:0) ⊔ k F ( k ) × Σ k X k (cid:1) / ∼ . Define F u ( c, x , · · · , x j ) = max i =1 , ··· ,j u ( x i ) for c ∈ F ( k ) and x i ∈ X . This is well-defined and G -equivariant. We check that F u satisfies Definition 4.17. For (ii), suppose we have g : X → X and y = ( c, x , · · · , x j ) ∈ F X with
F u ( y ) = max i =1 , ··· ,j u ( x i ) < . Then(
F u )( F g )( y ) = max i =1 , ··· ,j u ( gx i ) < . To check the conditions in Remark 4.18, we have
F u ( c, x , · · · , x j ) = 0 if and only if u ( x i ) = 0 for all i . This gives ( F u ) − (0) = F A ; F u ( c, x , · · · , x j ) < u ( x i ) < i . This gives ( F u ) − [0 , ⊂ F U ⊂ ( F h ) − ( F A ).For F = Map c ( M, − ), let F u ( f ) = max m ∈ M u ( f ( m )) for f ∈ Map c ( M, X ). Thisis well-defined since f is compactly supported. F u is G -equivariant since u is. Wecheck that F u satisfies Definition 4.17. For (ii), suppose we have g : X → X and f ∈ Map c ( M, X ) with
F u ( f ) = max m ∈ M u ( f ( m )) <
1. Then (
F u )( F g )( f ) = max m ∈ M u ( gf ( m )) <
1. For the conditions in Remark 4.18,
F u ( f ) = 0 if and only if u ( f ( m )) = 0 for all m ∈ M . This gives ( F u ) − (0) = Map c ( M, A ) =
F A ; F u ( f ) < u ( f ( m )) < m ∈ M . This gives ( F u ) − [0 , ⊂ F U ⊂ ( F h ) − ( F A ). The sameargument works for F = Map ∗ ( M + , − ).The functor F = Σ V can not be admissible in the sense of Definition 4.17, becausefor the pair ( X, A ) = ( S , pt) and any NDR representation ( h, u ) of it,( F h ) − ( F A ) = Σ V ( h − A ) QUIVARIANT FACTORIZATION HOMOLOGY 29 does not contain an open neighborhood of the base point of Σ V X , which leaves no roomfor U ′ to exist. Nevertheless, using the fact that ( S V , ∞ ) is an NDR pair, (Σ V X, Σ V A )is still an NDR pair by a based version of [May72, Lemma A.3]. (cid:3) Definition 4.20 (Lemma 1.9 of [MMOar]) . A simplicial G -space X • is Reedy cofibrantif all degeneracy operators s i are G -cofibrations.The following lemma shows that monadic bar constructions are Reedy cofibrant. Lemma 4.21 (adaptation of Proposition A.10 of [May72]) . Let C be a reduced operadin G -spaces such that the unit map η : pt → C (1) gives a non-degenerate base point.Let C be the reduced monad associated to C . Let A be a C -algebra in G Top ∗ and F : G Top ∗ → G Top ∗ be a right- C -module functor. Suppose that F sends NDR pairs toNDR pairs. Then B • ( F, C, A ) is Reedy cofibrant.Proof. We need to show that for any n ≥ ≤ i ≤ n , the degeneracy map s in = F C i η C n − i A : F C n A → F C n +1 A is a G -cofibration. Write X = C n − i A . By Lemma 4.19, C sends NDR pairs to NDRpairs. Start from the NDR pair ( A, pt) and apply this functor ( n − i ) times, we getan NDR pair ( C n − i A, pt) = ( X, pt). Together with the assumption that C (1) is non-degenerately based, we can show ( CX, X ) is an NDR pair where X is identified withthe image η X : X → CX (see the proof of [May72, A.10]). Applying C another i times and then F , we get the NDR pair (cid:0) F C i +1 X, F C i X (cid:1) = (cid:0) F C n +1 A, F C n A (cid:1) . Thus s in = F C i η X is a G -cofibration. (cid:3) Corollary 4.22.
Let
M, V, A be as in Theorem 4.8. Then the following are Reedycofibrant simplicial G -spaces: B • (D fr V M , D fr V V , A ) , Map c ( M, Σ V (D fr V V ) • A ) and Map ∗ ( M + , Σ V (D fr V V ) • A ) . Proof.
In Lemma 4.21, we take C = D fr V V and respectively F = D fr V M , F = Map c ( M, Σ V − )or F = Map ∗ ( M + , Σ V − ). By Lemma 4.19, each F does send NDR pairs to NDRpairs. (cid:3) Dimension.
We start with an introduction to G -CW complexes and equivariantdimensions following [May96, I.3]. A G -CW complex X is a union of G -spaces X n obtained by inductively gluing cells G/K × D n for subgroups K ⊂ G via G -maps alongtheir boundaries G/K × S n − to the previous skeleton X n − . Conventionally, X − = ∅ .We shall look at functions from the conjugacy classes of subgroups of G to Z ≥− andtypically denote such a function by ν . We say that a G -CW complex X has dimension ≤ ν if its cells of orbit type G/H all have dimensions ≤ ν ( H ), and that a G -space X is ν -connected if X H is ν ( H )-connected for all subgroups H ⊂ G , that is, π k ( X H ) = 0for k ≤ ν ( H ). We allow ν ( H ) = − X H = ∅ .For the purpose of induction in this paper, we use the following ad hoc definition: Definition 4.23.
A based G -CW complex is a union of G -spaces X n obtained byinductively gluing cells to X − = pt. We refer to the base point as ∗ . And we do NOTcount the point in X − as a cell for a based G -CW complex, excluding it from countingthe dimension as well. This is not the same as a based G -CW complex in [May96, Page18], where the base point is put in the 0-skeleton X . Fix a subgroup H ⊂ G . We have the double coset formula(4.24) G/K ∼ = a ≤ i ≤| H \ G/K | H/K i as H -sets,where each K i = H ∩ g i Kg − i for some element g i ∈ G . So a (based) G -CW structureon X restricts to a (based) H -CW structure on the H -space Res GH X . A function ν fromthe conjugacy classes of subgroups of G to Z ≥− induces a function from the conjugacyclasses of subgroups of H to Z ≥− , which we still call ν . However, for X of dimension ≤ ν , Res GH X may not be of dimension ≤ ν , as we see in (4.24) that an H/K i -cell cancome from a G/K -cell for a larger group K . For a function ν , we define the function d ν to be d ν ( K ) = max K ⊂ L ν ( L ) . Then Res GH X is of dimension ≤ d ν . Remark 4.25.
More specifically, one can define the dimension of a (based) G -CWcomplex X to be the minimum ν such that X is of dimension ≤ ν . Suppose X hasdimension ν . Then from (4.24), we get:(i) The (based) H -CW complex Res GH X has dimension µ , where µ ( K ) = max K ⊂ LK = L ∩ H ν ( L ) . We have µ ≤ d ν , and it can be strictly less. (For a trivial example, take H = G .)(ii) The (based) CW-complex X H has dimension µ ( H ) = d ν ( H ). (In the based case,we also exclude the base point from counting the dimension of X H .)We define the dimension of a representation V to be dim( V )( H ) = dim( V H ) for H representing a conjugacy class of subgroups of G . Note that d dim( V ) = dim( V ).The goal of this subsection is to give a sufficent condition for the following map(4.26) to be a weak G -equivalence. Let X be a finite based G -CW complex and K • bea simplicial G -space. Then the levelwise evaluation is a G -map | Map ∗ ( X, K • ) | ∧ X ∼ = | Map ∗ ( X, K • ) ∧ X | → | K • | , whose adjoint gives a G -map(4.26) | Map ∗ ( X, K • ) | → Map ∗ ( X, | K • | ) . Non-equivariantly, it is one of the key steps in May’s recognition principal [May72] torealize that (4.26) is a weak equivalence when the dimension of X is small comparedto the connectivity of K • . May proved this using quasi-fibrations, a concept that goesback to Dold–Thom. Equivariantly, one has a similar result (see Theorem 4.29). It isdue to Hauschild and written down by Costenoble–Waner [CW91]. Definition 4.27.
A map p : Y → W of spaces is a quasi-fibration if p is onto andit induces an isomorphism on homotopy groups π ∗ ( Y, p − ( w ) , y ) → π ∗ ( W, w ) for all w ∈ W and y ∈ p − ( w ). In other words, there is a long exact sequence on homotopygroups of the sequence p − ( w ) → Y → W for any w ∈ W .Usually, the geometric realization of a levelwise fibration is not a fibration. Thefollowing theorem gives conditions when it is a quasi-fibration, which is good enoughfor handling the homotopy groups. QUIVARIANT FACTORIZATION HOMOLOGY 31
Theorem 4.28. ( [May72, Theorem 12.7] ) Let p : E • → B • be a levelwise Hurewiczfibration of pointed simplicial spaces such that B • is Reedy cofibrant and B n is connectedfor all n . Set F • = p − ( ∗ ) . Then the realization | E • | → | B • | is a quasi-fibration withfiber | F • | . We need the following:
Theorem 4.29.
Let G be a finite group. If X is a finite based G -CW complex ofdimension ≤ ν and K • is a simplicial G -space such that for any n , K n is d ν -connected,then the natural map (4.26) | Map ∗ ( X, K • ) | → Map ∗ ( X, | K • | ) is a weak G -equivalence.Proof. Let ∗ = X − ⊂ X ⊂ X ⊂ · · · ⊂ X d ν ( e ) = X be the G -CW skeleton of X . Weuse induction on k to show that(i) Map ∗ ( X k , K n ) H is connected for all n and H ⊂ G .(ii) | Map ∗ ( X k , K • ) | H → Map ∗ ( X k , | K • | ) H is a weak equivalence for all H ⊂ G ;The base case k = − X k → X k +1 → X k +1 /X k and map it into K • . We then apply (4.26) and get the following commutative diagram:(4.30) | Map ∗ ( X k +1 /X k , K • ) | H | Map ∗ ( X k +1 , K • ) | H | Map ∗ ( X k , K • ) | H Map ∗ ( X k +1 /X k , | K • | ) H Map ∗ ( X k +1 , | K • | ) H Map ∗ ( X k , | K • | ) H Since maps out of a cofiber sequence form a fiber sequence, we have a fiber sequencein the second row and a realization of the following levelwise fiber sequence in the firstrow:(4.31) Map ∗ ( X k +1 /X k , K • ) H Map ∗ ( X k +1 , K • ) H Map ∗ ( X k , K • ) H By the inductive hypothesis (i) and Theorem 4.28, it realizes to a quasi-fibration.We first show the inductive case of (i). Suppose that we have X k +1 /X k = ∨ i ( G/K i ) + ∧ S k +1 , where { K i } i is a finite sequence of subgroups of G . This implies ν ( K i ) ≥ k + 1.From (4.24), we have X k +1 /X k ∼ = ∨ i ∨ j ( H/K i,j ) + ∧ S k +1 as a space with H -action,where each K i,j is G -conjugate to a subgroup of K i . Since d ν ( K i,j ) ≥ ν ( K i ), we have d ν ( K i,j ) ≥ k + 1 and the following space is connected by assumption:Map ∗ ( X k +1 /X k , K n ) H = Y i Map ∗ ( S k +1 , K K i,j n ) . This space is the fiber in (4.31). The connectedness of the base space by the inductivehypothesis (i) implies that of the total space.We next show the inductive case of (ii). Commuting geometric realization with finiteproduct and fixed point, the left vertical map of (4.30) is a product of maps | Map ∗ ( S k +1 , K K i,j • ) | → Map ∗ ( S k +1 , | K K i,j • | ) . These maps are weak equivalences by [May72, Theorem 12.3]. By the inductive hy-pothesis (ii), the right vertical map is a weak equivalence. Comparing the long exactsequences of homotopy groups, this implies that the middle vertical map is also a weakequivalence. (cid:3)
Remark 4.32.
Non-equivariantly, supposing that dim( X ) = m , Miller [Mil15, Cor2.22] observed that the theorem is also true if K n is only ( m − n ,since the only thing that fails in the proof is (i) for k = m . Equivariantly, one needs (i)to hold for k < d ν ( e ). So an equivariant stingy man can only relax the assumption to K Hn being min { d ν ( H ) , d ν ( e ) − } -connected for all n and H .Just as a remark, the unbased version of Theorem 4.29 is the following: Theorem 4.33. ( [CW91, Lemma 5.4] ) Let G be a finite group. If Y is a finite (unbased) G -CW complex and K • is a simplicial G -space such that for any n , K n is dim( Y ) -connected, then the natural map | Map(
Y, K • ) | → Map( Y, | K • | ) is a weak G -equivalence. Theorem 4.33 is a consequence of Theorem 4.29 by taking X = Y ⊔ {∗} and usingRemark 4.25. Note that by adopting the strange convention of the dimension of abased G -CW complex, the dimension of Y is the same as X . On the other hand, wehave the cofiber sequence S → X + → X for a based G -CW complex X as well as theidentification of Map ∗ ( X + , K • ) with Map( X, K • ). If K • is G -connected, we can use thequasi-fibration technique and take Y = X in Theorem 4.33 to deduce Theorem 4.29.But there are also cases to apply Theorem 4.29 where K • is not required to be G -connected, for example, when X = ( G/H ) + ∧ S n for H = G . So Theorem 4.29 isslightly finer than Theorem 4.33.Finally, we prepare the following results for the application of Theorem 4.29 in thesetting of nonabelian Poincar´e duality Theorem 4.8. We need G -CW structures on G -manifolds M , which exist by work of Illman: Theorem 4.34 (Theorem 3.6 of [Ill78]) . For a smooth G -manifold M and a closedsmooth G -submanifold N , there exists a smooth G -equivariant triangulation of ( M, N ) . Lemma 4.35.
Let M be a V -framed manifold and A be a G -connected space, then(1) M + has the homotopy type of a G -CW complex of dimension ≤ dim( V ) .(2) K n = Σ V (D fr V V ) n A is dim( V ) -connected.Proof. (1) Since M is a V -framed, the exponential maps give local coordinate chartsof M H as a (possibly empty) manifold of dimension dim( V H ). If M is compact wetake W = M , otherwise we take a compact manifold W with boundary such that M is diffeomorphic to the interior of W . By Theorem 4.34, ( W, ∂W ) has a G -equivarianttriangulation. It gives a relative G -CW structure on ( W, ∂W ) with relative cells of type
G/H of dimension ≤ dim( V H ). The quotient W/∂W gives the desired G -CW modelfor M + .(2) For any subgroup H ⊂ G , we have K Hn = (Σ V (D fr V V ) n A ) H = Σ V H ((D fr V V ) n A ) H .By Lemma 4.12, ((D fr V V ) n A ) H is connected. So K Hn is dim( V H )-connected. Thus, K n is dim( V )-connected. (cid:3) QUIVARIANT FACTORIZATION HOMOLOGY 33
Appendix A. A comparison of scanning maps
The scanning map studied in Section 4.1 is a key input to the Nonabelian Poincar´eduality theorem. In this section we compare our scanning map (4.3) to other construc-tions.
Notation A.1.
For a G -manifold M , Sph(T M ) is the fiberwise one-point compatifi-cation of the tangent bundle of M . It is a G -fiber bundle over M with based fiber S n ,where the base point in each fiber is the point at infinity.Non-equivariantly, people have used the name scanning map to refer to differentbut related constructions. In slogan, it is a map from the (fattened) configurationspaces of a manifold M to compactly defined sections of T M , or compactly supportedsections of Sph(T M ). McDuff [McD75] was probably the first to study the scanningmap for general manifolds. She thought of it as the field of the point charges and provedhomological stability properties of this map. In our case of T M ∼ = M × V , the situationis simpler and we have defined a scanning map in (4.4): s S : a k ≥ D fr V M ( k ) / Σ k → Map c ( M, S V ) . The left hand side is a model of the configuration space as justified in Corollary 3.28 (1);the right hand side is equivalent to the compactly supported sections of Sph(T M ) ∼ = M × S V .We are interested in the scanning maps of Manthorpe–Tillman and McDuff, both ofwhich can be made equivariant without pain. The following table is a summary of thenatural domains and codomains of each construction:scanning map domain codomainthis paper, s framed embeddings V to M maps M + to S V Manthorpe–Tillman, ˜ s MT embeddings V to M sections of Sph(T M )McDuff, ˜ s MD configuration of points of M sections of Sph(T M )In this section, we focus on the case of V -framed manifolds M . Then these maps haveequivalent domains and codomains. We will show in Proposition A.7 and Proposition A.10that: Theorem A.2.
The scanning maps s X , s MD X and s MT X are G -homotopic after the changeof domain. Notation A.3.
In the above and subsequent paragraphs, • We use the letter s for scanning maps without labels and s X for labels in X . • A tilde is put on s to denote when the codomain is the sections of Sph(T M ),that is, before composition with the framing. • A superscript is put on s to distinguish between the different authors in theliterature.A.1. Scanning map from tubular neighborhood.
Non-equivariantly, Manthorpe–Tillman [MT14, Section 3.1] gave a map γ + : (cid:0) a k ≥ Emb( ⊔ k R n , M ) × Σ k X k (cid:1) / ∼ → Section c ( M, Sph(T M ) ∧ M τ X ) . Here, Section c is the space of compactly supported sections; τ X is the constant parametrizedbase space X × M over M and Sph(T M ) ∧ M τ X is the fiberwise smashing of Sph(T M )with X . (To translate, take their M = ∅ , Y = W × X . Their E k ( M, π ) is Emb( ` k R n , M ) × Σ k X k , and their Γ( W \ M , W \ M, π ) is Section c ( M, Sph(T M ) ∧ M τ X ).)The key feature of their construction is to exploit the data of the tubular neighbor-hood, so a framing on M is not needed. For example, when k = 1, we start with anembedding f ∈ Emb( R n , M ) and want to define γ + ( f ), a compactly supported sectionof Sph(T M ). The image of f is a tubular neighborhood of the image of 0 ∈ V in M ,and f induces an inclusion of bundles df : T R n → T M . There is a canonical diagonalsection R n → R n × R n ∼ = T R n . Pushing this section by df gives γ + ( f ).We can modify their γ + by replacing R n by the representation V to get γ + V : Emb M ( X ) ≡ (cid:0) a k ≥ Emb( ⊔ k V, M ) × Σ k X k (cid:1) / ∼ → Section c ( M, Sph(T M ) ∧ M τ X ) . We then precompose with the forgetting map D fr V M ( X ) → Emb M ( X ) in Remark 3.8 toget(A.4) ˜ s MT X : D fr V M ( X ) → Section c ( M, Sph(T M ) ∧ M τ X ) . We describe how ˜ s MT X works on the subspace k = 1 and it is similar on the whole space.For the element ¯ f = ( f, α ) ∈ Emb fr V ( V, M ), we take the embedding f : V → M . Thederivative map of f is df : T V ∼ = V × V → T M . For each m ∈ image( f ), we need avector ˜ s MT ( f ) ∈ T m M that is determined by f . Denote v = f − ( m ) ∈ V . We have df v : V ∼ = T v V → T m M . Then the explicit formulas without or with labels are givenby(A.5) ˜ s MT ( ¯ f )( m ) = df v ( v ) and ˜ s MT X ( ¯ f , x )( m ) = df v ( v ) ∧ x. Both of them are G -maps.The V -framing φ M : T M → V induces Sph(T M ) ∧ M τ X ∼ = M × Σ V X . So we obtaina map which we still call the scanning map:(A.6) s MT X : D fr V M ( X ) → Map c ( M, Σ V X ) . A prior, this scanning map is different from the scanning map (4.2) in Section 4.1.For an element ¯ f = ( f, α ) where f : V → M with f ( v ) = m , we have s ( ¯ f )( m ) = v ∈ V in (4.2), while s MT ( ¯ f )( m ) = df v ( v ) ∈ T m M in (A.5). However, the data of a homotopyin defining the V -framed embedding ensure that the two approaches give homotopicscanning maps: Proposition A.7.
The map s X defined by (4.2) is G -homotopic to the map s MT X definedby (A.5).Proof. We show that s ≃ s MT : D fr V M ( k ) → Map c ( M, S V ). We write the homotopyexplicitly for k = 1 and the case for general k is similar. To unravel the data, anelement ¯ f = ( f, α ) ∈ D fr V M (1) consists of an embedding f : V → M and a homotopy α of two maps T V → V , where α (0) is the standard framing on V and α (1) is φ M ◦ df .The two scanning maps use the two endpoints of this homotopy. Namely, for m inImage( f ), write v = f − ( m ) ∈ V ∼ = T v V . Then the first approach can be written as s ( ¯ f )( m ) = v = α (0) v ( v ) QUIVARIANT FACTORIZATION HOMOLOGY 35 and the df -shifted-approach can be written as s MT ( ¯ f )( m ) = φ M df v ( v ) = α (1) v ( v ) . Now it is clear that we can define a homotopy H : D fr V M (1) × I → Map c ( M, S V ); H ( ¯ f , t )( m ) = α ( t ) f − ( m ) ( f − ( m )) . It is G -equivariant and gives a homotopy between H ( − ,
0) = s and H ( − ,
1) = s MT .The claim follows from observing that this homotopy is compatible with forgetting from k to k − (cid:3) A.2.
Scanning map using geodesic.
McDuff gave a geometric construction for F M ( S ) = a k ≥ F M ( k ) → Section c ( M, Sph(T M )) , Recall that F M ( k ) is the configuration space of k points in M . Note that the basepoint in each fiber of Sph(T M ) is the point at infinity; so such a compactly supportedsection of Sph(T M ) is just a vector field defined in the interior of a compact set on M that blows up to infinity towards the boundary.We first copy McDuff’s construction and fit it into a neat comparison with the pre-viously defined scanning maps.We focus on the case of M without boundary. Then we can translate her M ǫ to our M ; her E M can be identified with our Sph(T M ); her ˜ C M to our F M ( S ); her ˜ C ǫ ( M ) toa subspace of our Emb M ( S ).In summary, the scanning map goes in two steps: fatten up the configurations([McD75, Lemma 2.3]) and use geodesics to give compactly supported vector fields([McD75, p95]).(A.8) ˜ s MD : F M ( S ) ˜ C ǫ ( M ) Section c ( M, E M )Emb M ( S ) Section c ( M, Sph(T M )) fatten φ ǫ include η ∼ = γ + The commutative (A.8) is central in this section. In the first row, fatten and φ ǫ are thetwo steps in McDuff’s scanning map. The map γ + is from Section A.1. We will definethe undefined spaces and maps as we go along.Define˜ C ǫ ( M ) ≡ { exp m : T m M → M such that it is a diffeomorphism on the ǫ -ball } ;˜ C ǫ ( M ) ≡ { ( δ, e , · · · , e k ) | < δ ≤ ǫ, k ∈ N , e i ∈ ˜ C ǫ ( M ) for 1 ≤ i ≤ k, images of e i on the δ -balls are disjoint in M } . For preparation, we write down an explicit homeomorphism η ǫ : D ǫ ( R n ) → R n ; v tan (cid:0) π | v | ǫ (cid:1) v | v | . Here, D ǫ ( R n ) is the disk of radius ǫ in R n . Then, abusively we also have η : D (T m M ) /∂D (T m M ) ∼ = T m M ∪ {∞} ≡ Sph(T m M ) . Define E M to be the bundle over M whose fiber over m is D (T m M ) /∂D (T m M ), whichis identified with Sph(T m M ) through η . This is the right vertical map in (A.8).We give the vertical map in the middle of (A.8). For an element exp m ∈ exp m , thecomposite exp m ◦ η − ǫ is an embedding R n → M , so we can identify ˜ C ǫ ( M ) with asubspace of Emb( R n , M ). Similarly, we can include as subspace:˜ C ǫ ( M ) → Emb M ( S )( δ, e , · · · , e k ) ( e ◦ η − δ , · · · , e k ◦ η − δ )In McDuff’s first step, let us define φ ǫ and compare it to the map γ + locally. Puta Riemannian metric on M . The input for φ ǫ are the exponential maps in ˜ C ǫ ( M ) .Define φ ǫ (exp m )( m ) = ( ∗ if dist( m, m ) > ǫ ; dist( m,m ) ǫ · t ( m, m ) if dist( m, m ) ≤ ǫ. Here, the values are vectors in D (T m M ); t ( m, m ) is the unit tangent at m of theminimal geodesic from m to m ; dist( m, m ) is the distance between m and m . Now,it can be easily verified that γ + (exp m ◦ η − ǫ ) = η ◦ φ ǫ (exp m ) . We can work the same way to extend φ ǫ to ˜ C ǫ ( M ) and we have the commutativity partof (A.8): γ + ( e ◦ η − δ , · · · , e k ◦ η − δ ) = η ◦ φ ǫ ( δ, e , · · · , e k ) . In McDuff’s second step, we describe the fattening map in (A.8). We can take acontinuous positive function ǫ on M such that for any m ∈ M , the exponential mapexp m : T m M → M is always a diffeomorphism on the ǫ ( m )-ball. (We note the changehere: ǫ ( m ) is going to serve as the ǫ in the first step. It does not harm to think as if ǫ ( m ) = ǫ for all m .) Then, as is easily checked, we can choose a continuous positivefunction ¯ ǫ on F M ( S ) such that at any p = ( m , · · · , m k ) ∈ F M ( k ),(i) for all i = 1 , · · · , k , ¯ ǫ ( p ) ≤ ǫ ( m i ) ;(ii) the m i ’s are at least 2¯ ǫ ( p ) apart from each other.Tthe fattening map in (A.8) sends p = ( m , · · · , m k ) ∈ F M ( k ) to (¯ ǫ ( p ) , exp m , · · · , exp m k ) ∈ ˜ C ǫ ( M ). The continuity of ˜ s MD follows from the continuity of ¯ ǫ . Remark A.9.
The composite F M ( S ) ˜ C ǫ ( M ) Emb M ( S ) fatten include in (A.8) is up to homotopy the σ in (3.27).Equivariantly, we can take all of the Riemanian metric, ǫ and ¯ ǫ to be G -invariantbecause G is finite: for example, replacing ǫ by Σ g ∈ G ǫ ( g − ) / | G | will do. Then ˜ s MD defined by (A.8) is G -equivariant. We can fiberwise smash with labels to get˜ s MD X : F M ( X ) → Section c ( M, Sph(T M ) ∧ M τ X ) . We note that there is no V involved in ˜ s MD X . When M is V -framed, we can compose itwith the V -framing on M to get s MD X : F M ( X ) → Map c ( M, Σ V X ) . QUIVARIANT FACTORIZATION HOMOLOGY 37
This scanning map s MD X is good only for studying the configuration spaces, possiblywith labels. It depends on the fattening-up radius ¯ ǫ , which is not recorded explicitly inthe data. The choice does not matter because a different choice of the fattening-up willgive a homotopic scanning map. But for the purpose of a scanning map out of “config-uration spaces with summable labels” or the factorization homology, remembering theradius is important to sum the labels.We have seen three scanning maps so far: s X in (4.2), s MT X in (A.5) and s MD X in (A.8).We have shown that s X and s MT X are G -homotopic in Proposition A.7. We compare s MD X and s MT X in the following proposition. Proposition A.10.
The following diagram is G -homotopy commutative: D fr V M X Map c ( M, Σ V X ) F M X s MT X ev s MD X Proof.
Recall that s MT X is the composite of the forgetting map and γ + V : s MT X : D fr V M X → Emb M ( X ) γ + V → Map c ( M, Σ V X ) . By (A.8) and Remark A.9, we have a homotopy commutative diagram:Emb M ( X ) Map c ( M, Σ V X ) F M ( X ) γ + V σ s MD X By Corollary 3.28(2), σ ◦ ev is G -homotopic to the forgetting map D fr V M X → Emb M ( X ).So the claim follows. (cid:3) A.3.
Scanning equivalence.
We are interested in when the scanning map is an equiv-alence. In this subsection, we list Rourke–Sanderson’s result in [RS00]. Their work isbased on McDuff’s scanning map. The C M X in their paper is our ( F M X ) G . Definition A.11.
Let C and C ′ be A ∞ - G -spaces. An A ∞ - G -map f : C → C ′ is calleda weak group completion if for any subgroup H ⊂ G , there is a homotopy equivalenceΩB( C H ) ≃ ( C ′ ) H and f H is homotopic to C H → ΩB( C H ) ≃ ( C ′ ) H .Note that when C is an A ∞ - G -space and H ⊂ G , the fixed point space C H is an A ∞ -space; so f H is up to homotopy a group completion of C H . Theorem A.12.
The scanning map s MD X : F M X → Map c ( M, Σ V X ) is:(1) a weak G -equivalence if X is G -connected,(2) or a weak group completion if V ∼ = W ⊕ and M ∼ = N × R . Here, W is a ( n − -dimension G -representation and N is a W -framed closed G -manifold,so that N × R is V -framed. Proof. (1) is [RS00, Theorem 5]. For (2), we first note that when M ∼ = N × R , the map s MD X factors in steps as: F M X = F R ( F N X ) → Map c ( R , Σ F N ( X ))(A.13) → Map c ( R , F N (Σ X ))(A.14) → Map c ( R , Map c ( N, Σ W X )) . (A.15)Here, (A.13) and (A.15) are scanning maps for manifolds R and N ; (A.14) sends anelement p ∧ t for a configuration p on N with labels in X and t ∈ S to the sameconfiguration on N with labels suspended all by t in Σ X . All spaces presented have A ∞ -structures from the factor R in M : for any space Y , both the labeled configurationspace F R Y and the mapping space Map c ( R , Y ) ≃ Ω Y have obvious A ∞ -structures.The map (A.15) is a weak G -equivalence by applying part (1) with M replaced by N and X replaced by Σ X , which is G -connected. It suffices to show the composite of(A.13) and (A.14), denoted as j , is a weak group completion.[RS00, Theorem 3] constructed a homotopy equivalence q : B (cid:0) ( F M X ) G (cid:1) ≃ (cid:0) F N (Σ X ) (cid:1) G . Moreover, in Page 548, they established a homotopy commutative diagram:( F M X ) G Map c ( R , (cid:0) F N (Σ X )) (cid:1) G Map c ( R , B (cid:0) ( F M X ) G (cid:1) ) Map c ( R , (cid:0) F N (Σ X ) (cid:1) G ) j G Ω q The left column is the group completion map for the A ∞ -space ( F M X ) G . Since q isa homotopy equivalence, j G is a weak group completion. This remains true for anysubgroup H ⊂ G replacing G . Therefore, j is a weak group completion. (cid:3) Remark A.16. [RS00] does not assume the manifold M to be framed. Without theframing on M , Theorem A.12 is true in the following form:The scanning map ˜ s MD X : F M X → Section c ( M, Sph(T M ) ∧ M τ X ) is(1) a weak G -equivalence if X is G -connected,(2) or a weak group completion if M ∼ = N × R . Appendix B. A comparison of θ -framed morphisms In Section 3.1, we defined the θ -framed embedding space of θ -framed bundles usingpaths in the θ -framing. In this appendix, we compare this approach to an alternativedefinition following Ayala–Francis [AF15, Definition 2.7] in Proposition B.10. With thisalternative definition, we identify the automorphism G -space Emb θ ( V, V ) of V in Mfld θ in Theorem B.14; the special case θ = fr V has been treated directly in Section 3.3.B.1. The θ -framed maps. The classification theorem says that isomorphism classes ofvector bundles are in bijection to homotopy classes of maps to a classifying space. Pass-ing to the classification maps seems to lose the information about morphisms betweenbundles, but it turns out not to. We show that the automorphism space of a bundle isequivalent to the space of homotopies of a chosen classifying map in Corollary B.9. Tothis end, we first define a suitable “over category up to homotopy”.
QUIVARIANT FACTORIZATION HOMOLOGY 39
Let B be a G -space. A typical example is to take B = B G O ( n ). Then we have a Top-enriched over category G Top / B : the objects are G -spaces over B , and the morphismsare G -maps over B . Explicitly, for G -spaces over B given by G -maps φ M : M → B and φ N : N → B , the space Hom G Top / B ( M, N ) is the pullback displayed in the followingdiagram: (note that we have Hom G Top = Map G )(B.1) Hom G Top / B ( M, N ) Map G ( M, N ) ∗ Map G ( M, B ) φ N ◦−{ φ M } Now we want to work with G -spaces over B up to homotopy. We modify the morphismspace by taking the homotopy pullback in (B.1). Just like the difference between G Topand Top G , we have two versions: the Top-enriched G Top h / B and the G Top-enrichedTop h G / B . That is, we have homotopy pullback diagrams of spaces in (B.2) and of G -spaces in (B.3):(B.2) Hom G Top h / B ( M, N ) Map G ( M, N ) ∗ Map G ( M, B ) φ N ◦−{ φ M } (B.3) Hom Top h G / B ( M, N ) Map(
M, N ) ∗ Map(
M, B ) φ N ◦−{ φ M } Using the Moore path space model for the homotopy fiber as given in the followingdefinition, one can define unital and associative compositions to make G Top h / B andTop h G / B categories. Definition B.4.
For φ M : M → B and φ N : N → B , the space Hom G Top h / B ( M, N )and the G -space Hom Top h G / B ( M, N ) are given by:Hom G Top h / B ( M, N ) = { ( f, α, l ) | f ∈ Map G ( M, N ) , α ∈ Map( R ≥ , Map G ( M, B )) ,l ∈ Map(Map G ( M, N ) , R ≥ ) such that l is locally constant ,α (0) = φ M , α ( t ) = φ N ◦ f for t ≥ l ( f ) } . Hom
Top h G / B ( M, N ) = { ( f, α, l ) | f ∈ Map(
M, N ) , α ∈ Map( R ≥ , Map(
M, B )) ,l ∈ Map(Map(
M, N ) , R ≥ ) such that l is locally constant ,α (0) = φ M , α ( t ) = φ N ◦ f for t ≥ l ( f ) } . Remark B.5.
Roughly speaking, a point in the morphism space G Top h / B is a G -map f ∈ Map G ( M, N ) and a G -homotopy from φ M to φ N ◦ f in the following diagram: NM B φ N φ M f A point in the morphism space Top h G / B is a map f ∈ Map(
M, N ) and a homotopy from φ M to φ N ◦ f ; the map f is not necessarily a G -map, but we do require φ M and φ N tobe G -maps. And we haveHom G Top h / B ( M, N ) ∼ = (Hom Top h G / B ( M, N )) G . The category Top h G / B models θ -framed bundles: Proposition B.6.
For i = 1 , , let E i → B i be G - n -vector bundles with θ -framings φ i : E i → θ ∗ ζ n . We have the following equivalences of G -spaces that are natural withrespect to the two variables as well as the tangential structure: β : Hom θ ( E , E ) ∼ −→ Hom
Top h G / B ( B , B ) . Proof.
One can restrict bundle maps to get maps on the base spaces. We denote thisrestriction map by π . From our definition of Hom θ in Definition 3.4 and Hom Top h G / B inDefinition B.4, π induces the map β and they fit in the following commutative diagramof G -spaces:(B.7) Hom θ ( E , E ) Hom Top h G / B ( B , B )Hom( E , E ) Map( B , B )Hom( E , θ ∗ ζ n ) Map( B , B ) β ∼ πφ ◦− y φ ◦− π We claim that the bottom square is a pullback. Since each column is a homotopy fibersequence, this implies immediately that β is a G -equivalence.To show the claim, first we note that the isomorphism φ : E ∼ = φ ∗ θ ∗ ζ n establishes E as a pullback of θ ∗ ζ n over φ . So a bundle map E → E is determined by a map onthe base f : B → B and a bundle map ( ¯ ϕ, ϕ ) : ( E , B ) → ( ζ n , B ) satisfying ϕ = φ f . E E θ ∗ ζ n B B B ¯ ϕ y f φ (cid:3) We remark that in Proposition B.6, π is not a homotopy equivalence to its image.In other words, a vector bundle map is not just a map on the bases. In contrast, a θ -framed vector bundle map can be seen as a map on the bases as β is an equivalence. QUIVARIANT FACTORIZATION HOMOLOGY 41
The “classical” bundle maps are the θ -framed bundle maps for the tangential struc-ture θ = id : B G O ( n ) → B G O ( n ): Lemma B.8.
For G -vector bundles E i → B i , i = 1 , , we have an equivalence of G -spaces: α : Hom id ( E , E ) ∼ −→ Hom( E , E ) . Proof.
By definition, Hom id ( E , E ) is the homotopy fiber of φ ◦ − , so we have ahomotopy fiber sequence of G -spaces:Hom id ( E , E ) Hom( E , E ) Hom( E , ζ n ) α φ ◦− . By Lemma 2.38, we know Hom( E , ζ n ) is G -contractible. So α is a G -equivalence. (cid:3) Corollary B.9.
For G -vector bundles E i → B i , i = 1 , , we have an equivalence of G -spaces: Hom( E , E ) ≃ Hom
Top h G / BGO ( n ) ( B , B ) . Proof.
This follows from Proposition B.6 and Lemma B.8. (cid:3)
Proposition B.10.
The G -space Emb θ ( M, N ) as defined in Definition 3.6 is the ho-motopy pullback displayed in the following diagram of G -spaces: (B.11) Emb θ ( M, N ) Hom
Top h G / B ( M, N )Emb(
M, N ) Hom
Top h G / BGO ( n ) ( M, N ) Proof.
The lower horizontal map in (B.11) is neither obvious nor canonical. We takeit as the composite in the following commutative diagram with a chosen G -homotopyinverse to α . The maps α and β are G -equivalences by Proposition B.6 and Lemma B.8.(B.12) Emb θ ( M, N ) Hom θ (T M, T N ) Hom Top h G / B ( M, N )Hom id (T M, T N ) Hom Top h G / BGO ( n ) ( M, N )Emb(
M, N ) Hom(T M, T N ) ∼ β ∼ β ∼ αd As defined in Definition 3.6, Emb θ ( M, N ) is the pullback in the left square. It is clearthat it is also equivalent to the homotopy pullback of the whole square. (cid:3)
We can take (B.11) as an alternative definition to (3.7). In practice, (3.7) is easierto deal with. First, the right vertical map in the square is a fibration so the diagram isan actual pullback. Second, the map d is easy to describe. On the other hand, (B.11)has a conceptual advantage. It can be viewed as a comparison of the θ -framing to thetrivial framing id : B G O ( n ) → B G O ( n ). B.2.
Automorphism space of ( V, φ ) . With this alternative description of θ -framedmapping spaces in Section B.1, we can identify the automorphism G -space Emb θ ( V, V )of V in Mfld θ by first identifying of the automorphism G -space Hom θ (T V, T V ) of T V in Vec θ . Notation B.13. As φ is an equivariant map, φ (0) for the origin 0 ∈ V is a G -fixedpoint in B . We denote by Λ φ B the Moore loop space of B at the base point φ (0). Theorem B.14.
We have the following:(1) There is an equivalence of monoids in G -spaces Hom θ (T V, T V ) ∼ → Λ φ B, which is natural with respect to tangential structures θ : B → B G O ( n ) . Here,the group G acts on both sides by conjugation.(2) The automorphism G -space Emb θ ( V, V ) of ( V, φ ) in Mfld θ fits in the followinghomotopy pullback diagram of G -spaces: Emb θ ( V, V ) Λ φ B Emb(
V, V ) O ( V ) d Proof. (1) We have Hom
Top h G / B ( V, V ) from Definition B.4 and showed in Proposition B.6that restriction-to-the-base gives a natural G -equivalence: β : Hom θ (T V, T V ) ∼ → Hom
Top h G / B ( V, V ) . Let pt be the G -space over B given by φ (0) : pt → B . We claim that the two mapsinc : 0 → V and proj : V → pt can be lifted to give equivalences of V ≃ pt in Top h G / B .If so, pre-composing with inc and post-composing with proj giveHom Top h G / B ( V, V ) ∼ → Hom
Top h G / B (pt , pt) ∼ = Λ φ B. It remains to verify the claim, which is a routine job. We choose the lifts of inc andproj given by I = (inc , α , ∈ Hom
Top h G / B (pt , V ) , where α ( t ) = φ (0) for all t ≥ .P = (proj , α , ∈ Hom
Top h G / B ( V, pt) , where α ( t ) = ( φ ◦ h t , ≤ t < φ (0) , t ≥ h t : V → V is any chosen homotopy from h = id to h = proj. Then we havean obvious homotopy: P ◦ I = (id , const φ (0) , ≃ (id , const φ (0) ,
0) = id pt and using the contraction h t , we can also construct a homotopy: I ◦ P = (proj , α , ≃ (id , const φ ,
0) = id V . (cid:3) (2) This is an assembly of part (1), Proposition B.10 and Theorem 2.35. However,we note that the map Λ φ B → O ( V ) is only a non-canonical G -equivalence. The authordoes not know how to upgrade it to a map of G -monoids. So although all spaces QUIVARIANT FACTORIZATION HOMOLOGY 43 displayed in the pullback diagram are G -monoids, it is not obvious whether one canwrite Emb θ ( V, V ) as a pullback of G -monoids.To be more precise, we show how the quoted results assemble. We have the followinglarge commutative diagram (B.15) expanding (B.12). Note that this is a commutativediagram of G -monoids. (B.15)Emb θ ( V, V ) Hom θ (T V, T V ) Hom Top h G / B ( V, V )Hom id (T V, T V ) Hom Top h G / BGO ( n ) ( V, V ) Λ φ B Emb(
V, V ) Hom(T V, T V ) Hom id ( V, V ) Λ φ B G O ( n )Hom( V, V ) = O ( V ) ∼ β ○ ∼∼ β ∼ α ∼ ∼ ○ ○ ∼ ○ ∼ β ∼ α The map α is studied in Lemma B.8. The map β and the square 1 ○ are in Proposition B.6.The diagonal unlabeled maps are all induced by the inclusion V → T V and the pro-jection T V → V . In particularly, the parallelogram 2 ○ is in part (1). Naturality of α and β gives the commutativity of 3 ○ and 4 ○ . Now, d in the theorem is the compositeEmb( V, V ) Hom(T V, T V ) Hom( V, V ) . d ∼ It can be seen that the vertical map in the theorem involves choosing an inverse of the β displayed in the third line. Remark B.16.
From Remark 2.36, we have a zigzag of equivalences of G -monoids forany b in the V -indexed component of ( B G O ( n )) G : Λ b B G O ( n ) ( e Λ b E G O ( n )) / Π O ( V ) . ξ ∼ ψ ∼ This zigzag is hidden in (B.15). Recall that we abusively use φ to denote both T V → ζ n and V → B G O ( n ). Firstly, b = φ (0) ∈ B G O ( n ) is a point in the desired component,and we have Hom id ( V, V ) ∼ = ( e Λ b E G O ( n )) / Π . This is because Hom(
V, ζ n ) = Fr V ( B G O ( n )) ∼ = E G O ( n ) (see Definition 2.24 for Fr V ),and the framing φ (0) ∈ Hom(
V, ζ n ) corresponds to a chosen point z ∈ E G O ( n ) over b .The point z is G -fixed using the G -action on Fr V ( B G O ( n )). We can identify the pathdata of an element of Hom id ( V, V ), as defined in Definition 3.4, to the path data of arepresentative element of ( e Λ b E G O ( n )) / Π that starts at z , as described in Theorem 2.35.Secondly, the maps ψ and ξ are just the maps α and β . In other words, (2.37) canidentified with the following part of (B.15): Λ φ B G O ( n ) Hom id ( V, V ) Hom(
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