aa r X i v : . [ m a t h . A T ] J a n AN INTRODUCTION TO THE CATEGORY OF SPECTRA
N. P. STRICKLAND
These notes give a brief introduction to the category of spectra as defined in stable homotopy theory. Inparticular, Section 5 discusses an extensive list of examples of spectra whose properties have been found tobe interesting. Although many references are given, most of them are old. A survey of more recent researchand exposition would be valuable, but is not attempted here.1.
Introduction
Early in the history of homotopy theory, people noticed a number of phenomena suggesting that it wouldbe convenient to work in a context where one could make sense of negative-dimensional spheres. Let X bea finite pointed simplicial complex; some of the relevant phenomena are as follows. • For most n , the homotopy sets π n X are abelian groups. The proof involves consideration of S n − and so breaks down for n <
2; this would be corrected if we had negative spheres. • Calculation of homology groups is made much easier by the existence of the suspension isomorphism e H n + k Σ k X = e H n X . This does not generally work for homotopy groups. However, a theorem ofFreudenthal says that if X is a finite complex, we at least have a suspension isomorphism π n + k Σ k X = π n + k +1 Σ k +1 X for large k . If could work in a context where S − k makes sense, we could smasheverything with S − k to get a suspension isomorphism in homotopy parallel to the one in homology. • We can embed X in S k +1 for large k , and let Y be the complement. Alexander duality says that e H n Y = e H k − n X , showing that X can be “turned upside-down”, in a suitable sense. The shift by k is unpleasant, because the choice of k is not canonical, and the minimum possible k depends on X .Moreover, it is unsatisfactory that the homotopy type of Y is not determined by that of X (evenafter taking account of k ). In a context where negative spheres exist, one can define DX = S − k ∧ Y ;one finds that e H n DX = e H − n X and that DX is a well-defined functor of X , in a suitable sense. • The Bott periodicity theorem says that the homotopy groups of the infinite orthogonal group O ( ∞ )satisfy π k +8 O ( ∞ ) = π k O ( ∞ ) for all k ≥
0. It would be pleasant and natural to extend this patternto negative values of k , which would again require negative spheres.Considerations such as these led to the construction of the Spanier-Whitehead category F of finite spectra,which we briefly survey in Section 2. Although fairly straightforward, and very beautiful and interesting,this category has two defects. • Many of the most important examples in homotopy theory are infinite complexes: Eilenberg-MacLane spaces, classifying spaces of finite groups, infinite-dimensional grassmannians and so on.The category F is strongly tied to finite complexes, so a wider framework is needed to capture theseexamples. • Ordinary homotopy theory is made both easier and more interesting by its connections with geometry.However, F is essentially a homotopical category, with no geometric structure behind it. This alsoprevents a good theory of spectra with a group action, or of bundles of spectra over a space, or ofdiagrams of spectra.The first problem was addressed by a number of people, but the definitive answer was provided by Boardman.He constructed a category B with excellent formal properties parallel to those of F , whose subcategory offinite objects (suitably defined) is equivalent to F . A popular exposition of this category is in Adams’book [1]. Margolis [39] gave a list of the main formal properties of B and its relationship with F . Heconjectured (with good evidence) that they characterise B up to equivalence. See [12] for some new evidencefor this conjecture, and [24] for an investigation of some related systems of axioms.The second problem took much longer to resolve. There have been a number of constructions of topologicalcategories whose associated homotopy category (suitably defined) is equivalent to B , with steadily improving ormal properties [17, 18, 25, 34]. There is also a theorem of Lewis [33] which shows that it is impossible tohave all the good properties that one might naively hope for. We will sketch one construction in Section 4.2. The finite stable category
Basics.
We first recall some basic definitions. In this section all spaces are assumed to be finite CWcomplexes with basepoints. (We could equally well use simplicial complexes instead, at the price of havingto subdivide and simplicially approximate from time to time.) We write 0 for all basepoints, and we write[
A, B ] for the set of based homotopy classes of maps from A to B . We define A ∨ B to be the quotient of thedisjoint union of A and B in which the two basepoints are identified together. We also define A ∧ B to be thequotient of A × B in which the subspace A × ∪ × B is identified with the single point (0 , A and B ; note that it is commutative and associative up to isomorphism and that S ∧ A = A , where S = { , } . Moreover, A ∧ ( B ∨ C ) = ( A ∧ B ) ∨ ( A ∧ C ) and [ A ∨ B, C ] = [
A, C ] × [ B, C ](so ∨ is the coproduct in the homotopy category of pointed spaces). We let S denote the quotient of [0 , A = S ∧ A . One can check that Σ sends the ball B n to B n +1 and the sphere S n to S n +1 , and the reduced homology of Σ A is just e H n (Σ A ) = e H n − ( A ). Thus, we thinkof Σ as shifting all dimensions by one.The quotient space of S in which 1 / S ∨ S , sowe get a map δ : S −→ S ∨ S and thus a map δ : Σ A −→ Σ A ∨ Σ A . It is well-known that the inducedmap δ ∗ : [Σ A, B ] × [Σ A, B ] −→ [Σ A, B ] makes [Σ
A, B ] into a group. There are apparently n different groupstructures on [Σ n A, B ], but it is also well-known that they are all the same, and they are commutative when n >
1. We have an evident sequence of maps[
A, B ] Σ −→ [Σ A, Σ B ] Σ −→ [Σ A, Σ B ] −→ . . . . Apart from the first two terms, it is a sequence of Abelian groups and homomorphisms. By a fundamentaltheorem of Freudenthal, after a finite number of terms, it becomes a sequence of isomorphisms. We define[Σ ∞ A, Σ ∞ B ] to be the group [Σ N A, Σ N B ] for large N , or if you prefer the colimit lim −→ N [Σ N A, Σ N B ]. Afterdoing a little point-set topology, one concludes that this is the same as the set [ A, QB ], where QB =lim −→ N Ω N Σ N B and Ω N C means the space of based continuous maps S N −→ C , with a suitable topology.2.2. Finite spectra.
One can define a category with one object called Σ ∞ A for each finite CW complex A , and morphisms [Σ ∞ A, Σ ∞ B ]. It is easy to see that Σ induces a full and faithful endofunctor of thiscategory. We prefer to arrange things so that Σ is actually an equivalence of categories. Accordingly, wedefine a category F whose objects are expressions of the form Σ ∞ + n A where A is a finite CW complex and n is an integer. (If you prefer, you can take the objects to be pairs ( n, A ).) We refer to these objects as finitespectra. The maps are [Σ ∞ + n A, Σ ∞ + m B ] = lim −→ N [Σ N + n A, Σ N + m B ] . Freudenthal’s theorem again assures us that the limit is attained at a finite stage. The functor Σ inducesa self-equivalence of the category F . There are evident extensions of the functors ∨ and ∧ to F such thatΣ ∞ A ∨ Σ ∞ B = Σ ∞ ( A ∨ B ) and Σ ∞ A ∧ Σ ∞ B = Σ ∞ ( A ∧ B ) (although care is needed with signs whendefining the smash product of morphisms). The category F is additive, with biproduct given by the functor ∨ . The morphism sets [ X, Y ] in F are finitely generated Abelian groups. One can define homology of finitespectra by H n Σ ∞ + m A = e H n − m A , and then the map H ∗ : Q ⊗ [ X, Y ] −→ Y n Hom( H n ( X ; Q ) , H n ( Y ; Q ))is an isomorphism. The groups [ X, Y ] themselves are known to be recursively computable, but the guaranteedalgorithms are of totally infeasible complexity. Nonetheless, there are methods of computation which requiremore intelligence than the algorithms but have a reasonable chance of success. .3. Stable homotopy groups of spheres.
Even the groups π Sn = [Σ ∞ + n S , Σ ∞ S ] are hard, and areonly known for n ≤
60 or so (they are zero when n < π S = Z { ι } π S = Z / { η } π S = Z / { η } π S = Z / { ν } π S = 0 π S = 0 π S = Z / { ν } π S = Z / { σ } Here ι is the identity map, and η comes from the map η : S = { ( z, w ) ∈ C | | z | + | w | = 1 } −→ C ∪ {∞} = S defined by η ( z, w ) = z/w . Similarly, ν comes from division of quaternions, and σ from division of octonions(but most later groups cannot be described in a similarly explicit way). The expression η really means η ◦ (Σ η ), and ν means ν ◦ (Σ ν ).Many general results are also known. For example, for any prime p , the p -torsion part of π Sn is known for n < p − p and is zero for n < p − n = 0). Both the rank and the exponent are finite butunbounded as n tends to infinity. The group π S ∗ is a graded ring, and is commutative in the graded sense.An important theorem of Nishida says that all elements of degree greater than zero are nilpotent.2.4. Triangulation.
The category F is not Abelian. Instead, it has a triangulated structure. This meansthat there is a distinguished class of diagrams of the shape X f −→ Y g −→ Z h −→ Σ X (called exact triangles)with certain properties to be listed below. In our case the exact triangles can be described as follows. Let A be a subcomplex of a finite CW complex B , and let C be obtained from B by attaching a cone I ∧ A along the subspace { } × A = A . There is an evident copy of B in C , and if we collapse it to a pointwe get a copy of Σ A . We thus have a diagram of spaces A −→ B −→ C −→ Σ A . We say that a diagram X −→ Y −→ Z −→ Σ X of finite spectra is an exact triangle if it is isomorphic to a diagram of the formΣ ∞ + n A −→ Σ ∞ + n B −→ Σ ∞ + n C −→ Σ ∞ + n +1 A for some n ∈ Z and some A , B and C as above. Incidentally,one can show that C is homotopy equivalent to the space B/A obtained from B by identifying A with thebasepoint.The axioms for a triangulated category are as follows. In our case, they all follow from the theory ofPuppe sequences in unstable homotopy theory.(a) Any diagram isomorphic to an exact triangle is an exact triangle.(b) Any diagram of the form 0 −→ X −→ X −→ Σ0 = 0 is an exact triangle.(c) Any diagram X f −→ Y g −→ Z h −→ Σ X is an exact triangle if and only if the diagram Y g −→ Z h −→ Σ X − Σ f −−−→ Σ X is an exact triangle.(d) For any map f : X −→ Y there exists a spectrum Z and maps g, h such that X f −→ Y g −→ Y h −→ Σ X isan exact triangle.(e) Suppose we have a diagram as shown below (with h missing), in which the rows are exact trianglesand the rectangles commute. Then there exists a (nonunique) map h making the whole diagramcommutative. U / / f (cid:15) (cid:15) V / / g (cid:15) (cid:15) W / / h (cid:15) (cid:15) Σ U Σ f (cid:15) (cid:15) X / / Y / / Z / / Σ X (f) Suppose we have maps X v −→ Y u −→ Z , and exact triangles ( X, Y, U ), (
X, Z, V ) and (
Y, Z, W ) asshown in the diagram. (A circled arrow U −→◦ X means a map U −→ Σ X .) Then there exist maps r and s as shown, making ( U, V, W ) into an exact triangle, such that the following commutativitieshold: au = rd es = (Σ v ) b sa = f br = c s (cid:17) (cid:17) b (cid:13) ⑦⑦⑦ ~ ~ ⑦⑦⑦ X uv / / v ❅❅❅❅❅❅❅❅ Z a _ _ ❅❅❅❅❅❅❅❅ f ❆❆❆❆❆❆❆❆ U r c (cid:13) ⑦⑦⑦ > > ⑦⑦⑦ Y d o o u ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ W e (cid:13) o o The last axiom is called the octahedral axiom (the diagram can be turned into an octahedron by lifting theouter vertices and drawing an extra line from W to U ). In our case it basically just says that when we haveinclusions A ⊆ B ⊆ C of CW complexes we have ( C/A ) / ( B/A ) =
C/B .One of the most important consequences of the axioms is that whenever X −→ Y −→ Z −→ Σ X is an exacttriangle and W is a finite spectrum, we have long exact sequences . . . −→ [ W, Σ − Z ] −→ [ W, X ] −→ [ W, Y ] −→ [ W, Z ] −→ [ W, Σ X ] −→ . . . and . . . ←− [Σ − Z, W ] ←− [ X, W ] ←− [ Y, W ] ←− [ Z, W ] ←− [Σ X, W ] ←− . . . . Thom spectra.
Let X be a finite CW complex, and let V be a vector bundle over X . The Thom space X V can be defined as the one-point compactification of the total space of V . This has many interestingproperties, not least of which is the fact that when V is an oriented bundle of dimension n , the reducedcohomology e H ∗ ( X V ) is a free module over H ∗ ( X ) on one generator in dimension n . This construction canbe generalised to virtual bundles, in other words formal expressions of the form V − W , except that we nowhave a Thom spectrum X V − W rather than a Thom space. The construction is to choose a surjective mapfrom a trivial bundle R n × X onto W , with kernel U say, and define X V − W = Σ ∞− n X V ⊕ U .2.6. Duality.
For any finite spectrum X , there is an essentially unique spectrum DX (called the Spanier-Whitehead dual of X ) equipped with a natural isomorphism [ W ∧ X, Y ] = [
W, DX ∧ Y ]. This can beconstructed in a number of different ways. One way is to start with a simplicial complex A and embed itsimplicially as a proper subcomplex of S N +1 for some N >
0. One can show that the complement of A has a deformation retract B which is a finite simplicial complex, and D Σ ∞ + n A = Σ ∞− n − N B . Note thatAlexander duality implies that H m X = H − m DX .An important example arises when X = Σ ∞ M + for some smooth manifold M , with tangent bundle τ say.It is not hard to show geometrically that D (Σ ∞ M + ) is M − τ , the Thom spectrum of the virtual bundle − τ over M ; this phenomenon is called Atiyah duality .We also write F ( X, Y ) = DX ∧ Y . This is a functor in both variables, it preserves cofibrations up to sign,and the defining property of DX can be rewritten as [ W, F ( X, Y )] = [ W ∧ X, Y ].2.7.
Splittings.
It often happens that we have a finite complex X that cannot be split into simpler pieces,but that the finite spectrum Σ ∞ X does have a splitting. Group actions are one fruitful source of splittings. Ifa finite group G acts on X , then the map G −→ Aut(Σ ∞ X ) extends to a ring map Z [ G ] −→ End(Σ ∞ X ). If e H ∗ X is a p -torsion group, then this will factor through ( Z /p n )[ G ] for large n . Any idempotent element in ( Z /p )[ G ]can be lifted uniquely to an idempotent in ( Z /p n )[ G ], which will give an idempotent in End(Σ ∞ X ) and thusa splitting of X . The methods of modular representation theory give good information about idempotentsin group rings, and thus a supply of interesting splittings. The Steinberg idempotent in ( Z /p )[ GL n ( Z /p )]gives particularly important examples [42, 44], as do various idempotents in ( Z /p )[Σ n ] (see [51, AppendixC]).Another common situation is to have a finite complex X and a filtration F X ⊆ F X ⊆ . . . ⊆ F n X = X that splits stably, giving an equivalence Σ ∞ X ≃ W n Σ ∞ F n X/F n − X of finite spectra. For example, one cantake X = U ( m ), and let F n X be the space of matrices A ∈ U ( m ) for which the rank of A − I is at most n .A theorem of Miller [40] says that the filtration splits stably, and that the quotient F n U ( m ) /F n − U ( m ) isthe Thom space of a certain bundle over the Grassmannian of n -planes in C m . Later we will explain how tointerpret Σ ∞ X when X is an infinite complex; there are many examples in which X has a stably split filtrationin which the quotients are finite spectra. This holds for X = BU ( n ) or X = Ω U ( n ) or X = Ω n S n + m , for xample. The splitting of Ω n S n + m is due to Snaith [54]; the Snaith summands in Ω S are called Brown-Gitler spectra , and they have interesting homological properties with many applications [8, 27, 52].3.
Cobordism and Morava K -theory We next outline the theory of complex cobordism [49], and the results of Hopkins, Devinatz and Smithshowing how complex cobordism reveals an important part of the structure of F .Given a space X and an integer n ≥ geometric n -chain in X to be a compact smooth n -manifold M (possibly with boundary) equipped with a continuous map f : M −→ X . We regard ( M , f )and ( M , f ) as equivalent if there is a differomorphism g : M −→ M with f g = f . We write GC n X for the set of equivalence classes, which is a commutative semigroup under disjoint union. We define adifferential ∂ : GC n X −→ GC n − X by ∂ [ M, f ] = [ ∂M, f | ∂M ]. One can make sense of the homology M O ∗ X = H ( GC ∗ X, ∂ ), and (because ∂ ( M × I ) = M ∐ M ) one finds that M O ∗ X is a vector space over Z /
2. In thecase where X is a point, one can use cartesian products to make M O ∗ (point) into a graded ring, whichis completely described by a remarkable theorem of Thom: it is a polynomial algebra over Z / x n in degree n for each integer n > k −
1. New perspectives on this answerand the underlying algebra were provided by Quillen [46, 47] and Mitchell [43]. One can also show that
M O ∗ X = M O ∗ (point) ⊗ Z / H ∗ ( X ; Z / M O ∗ X when X is a finite spectrum.The story changes however, if we work with oriented manifolds. This gives groups M SO ∗ X with a richerstructure; in particular, they are not annihilated by 2. There are various hints that complex manifolds wouldgive still more interesting invariants, but there are technical problems, not least the lack of a good theory ofcomplex manifolds with boundary. It turns out to be appropriate to generalize and consider manifolds with aspecified complex structure on the stable normal bundle, known as “stably complex manifolds”. The precisedefinitions are delicate; details are explained in [46], and Buchstaber and Ray [9] have provided naturallyoccuring examples where the details are important. In any case, one ends up with a ring M U ∗ = M U ∗ (point),and groups M U ∗ X for all spaces X that are modules over it. It is not the case that M U ∗ X = M U ∗ ⊗ Z H ∗ X ,but there is still a suspension isomorphism, which allows one to define M U ∗ X when X is a finite spectrum.One finds that this is a generalized homology theory (known as complex cobordism ), so it converts cofibreseqences of spectra to long exact sequences of modules. The nilpotence theory of Devinatz, Hopkins andSmith [13, 22, 51] (which will be outlined below) shows that M U ∗ X is an extremely powerful invariant of X .The ring M U ∗ turns out to be a polynomial algebra over Z with one generator in each positive evendegree. There is no canonical system of generators, but nonetheless, Quillen showed that M U ∗ is canonicallyisomorphic to an algebraically defined object: Lazard’s classifying ring for formal group laws [1, 49]. Thiswas the start of an extensive relationship between stable homotopy and formal group theory. The algebraprovides many natural examples of graded rings A ∗ equipped with a formal group law and thus a map M U ∗ −→ A ∗ . It is natural to ask whether there is a generalised homology theory A ∗ ( X ) whose value ona point is the ring A ∗ . Satisfactory answers for a broad class of rings A ∗ are given in [56], which surveysand consolidates a great deal of much older literature and extends newer ideas from [17]. In particular, wecan consider the rings K ( p, n ) ∗ = F p [ v n , v − n ], where p is prime and n > v n has degree 2( p n − M U ∗ using a well-known formal group law, and by old or new methods,one can construct an associated generalised homology theory K ( p, n ) ∗ X , known as Morava K -theory . It isconvenient to extend the definition by putting K ( p, ∗ X = H ∗ ( X ; Q ).A key theorem of Hopkins, Devinatz and Smith says that if f : Σ d X −→ X is a self-map of a finite spectrum X , and K ( p, n ) ∗ ( f ) = 0 for all primes p and all n ≥
0, then the iterated composite f m : Σ md X −→ X is zerofor large m , or in other words f is composition-nilpotent. They also show that Morava K -theory detectsnilpotence in a number of other senses, and give formulations involving the single theory M U instead of thecollection of theories K ( p, n ).If R is a commutative ring, it is well-known that the ideal of nilpotent elements is the intersection ofall the prime ideals, so the Zariski spectrum is unchanged if we take the quotient by this ideal. One candeduce that the classification of certain types of subcategories of the abelian category of R -modules is againinsensitive to the ideal of nilpotents. Our category F of finite spectra is triangulated rather than abelian,but nonetheless Hopkins and Smith developed an analogous theory and deduced a classification of the thicksubcategories of F , with many important consequences. . Boardman’s category B It is clearly desirable to have a category C analogous to F but without finiteness conditions. There arevarious obvious candidates: one could take the Ind-completion of F , or just follow the definition of F but allowinfinite CW complexes instead of finite ones. Unfortunately, these categories turn out to have unsatisfactorytechnical properties. The requirements were first assembled in axiomatic form by Margolis [39]; in outline,they are as follows: • C should be a triangulated category • Every family { X α } of objects in C should have a coproduct, written W α X α • For any
X, Y ∈ C there should be functorially associated objects X ∧ Y and F ( X, Y ) making C aclosed symmetric monoidal category. • If we let small( C ) be the subcategory of objects W for which the natural map L α [ W, X α ] −→ [ W, W α X α ] is always an isomorphism, then small( C ) should be equivalent to F .Historically, the work of Margolis came after Boardman’s construction of a category B satisfying the axioms,and Adams’s explanation [1] of a slightly different way to approach the construction. Margolis conjecturedthat if C satisfies the axioms then C is equivalent to B . Schwede and Shipley [53] have proved that this istrue, provided that C is the homotopy category of a closed model category in the sense of Quillen [16, 48]satisfying suitable axioms. There is also good evidence for the conjecture without this additional assumption.The objects of B are generally called spectra , although in some contexts one introduces different words todistinguish between objects in different underlying geometric categories.Probably the best approach to constructing B is via the theory of orthogonal spectra , as we now de-scribe [35]. Let V denote the category of finite-dimensional vector spaces over R equipped with an innerproduct. The morphisms are linear isomorphisms that preserve inner products. For any V ∈ V (withdim( V ) = n say) we write S V for the one-point compactification of V ; this is homeomorphic to S n . An or-thogonal spectrum X consists of a functor V → { based spaces } together with maps S U ∧ X ( V ) → X ( U ⊕ V )satisfying various continuity and compatibility conditions that we will not spell out. We write S for thecategory of orthogonal spectra. For example, a based space A gives an orthogonal spectrum Σ ∞ A with(Σ ∞ A )( U ) = S U ∧ A . A vector space T ∈ V gives positive and negative sphere spectra S ± T : the value ofthe sphere spectrum S T at U ∈ V is the sphere space S U ∧ S T = S U ⊕ T , and S − T ( U ) = { ( α, u ) | α ∈ V ( T, U ) , u ∈ U, h u, α ( T ) i = 0 } ∪ {∞} . For orthogonal spectra X and Y , the morphism set S ( X, Y ) has a natural topology, and we could define anassociated homotopy category by the rule [
X, Y ] = π S ( X, Y ). Unfortunately, the resulting category is notthe one that we want. Instead, we define the homotopy groups of X by the rule π k ( X ) = lim −→ N π k + N ( X ( R N )).We then say that a map f : X → Y is a weak equivalence if π ∗ ( f ) : π ∗ ( X ) → π ∗ ( Y ) is an isomorphism. We nowconstruct a new category Ho( S ) by starting with S and adjoining formal inverses for all weak equivalences.It can be shown that this is equivalent to B (or can be taken as the definition of B ).This process of adjoining formal inverses can be subtle. To manage the subtleties, we need the theory ofmodel categories in the sense of Quillen [16, 23]. In particular, this will show that Ho( S )( X, Y ) = π S ( X, Y )for certain classes of spectra X and Y ; this is enough to get started with computations and prove thatMargolis’s axioms are satisfied.Given orthogonal spectra X , Y and Z , a pairing from X and Y to Z consists of maps α U,V : X ( U ) ∧ Y ( V ) → Z ( U ⊕ V ) satisfying some obvious compatibility conditions. One can show that there is an orthogonalspectrum X ∧ Y such that pairings from X and Y to Z biject with morphisms from X ∧ Y to Z . Basicexamples are that S ∧ X = X and Σ ∞ A ∧ Σ ∞ B = Σ ∞ ( A ∧ B ) and S U ∧ S V = S U ⊕ V and S − U ∧ S − V = S − U − V .There is a natural map S U ∧ S − U → S that is a weak equivalence but not an isomorphism.This construction gives a symmetric monoidal structure on S . This in turn gives rise to a symmetricmonoidal structure on Ho( S ); however, there are some hidden subtleties in this step, which again are besthandled by the general theory of model categories. One consequence is that the topology of the classifyingspace of the symmetric group Σ k is mixed in to the structure of the k -fold smash product X ( k ) = X ∧ · · · ∧ X and the quotient X ( k ) / Σ k . ring spectrum is an object R ∈ Ho( S ) equipped with a unit map η : S → R and a multiplication map µ : R ∧ R → R such that the following diagrams in Ho( S ) commute: R ∧ R ∧ R µ ∧ / / ∧ µ (cid:15) (cid:15) R ∧ R µ (cid:15) (cid:15) R η ∧ / / " " ❋❋❋❋❋❋❋❋❋ R ∧ R µ (cid:15) (cid:15) R ∧ η o o | | ①①①①①①①①① R ∧ R µ / / R R
Because we now have a good underlying geometric category S , we can formulate a more precise notion: a strict ring spectrum is an object R ∈ S equipped with morphisms S η −→ R µ ←− R ∧ R such that the abovediagrams commute in S (not just in Ho( S )).The symmetric monoidal structure on S includes a natural map τ XY : X ∧ Y → Y ∧ X . We say thata strict ring spectrum R is strictly commutative if µ ◦ τ RR = µ . This is a surprisingly stringent condition,with extensive computational consequences. A key point is that we have an iterated multiplication map R ( k ) / Σ k → R , and this brings into play the structure of R ∗ ( B Σ k ).5. Examples of spectra
Some important functors that construct objects of B are as follows:(a) For any based space X there is a suspension spectrum Σ ∞ X ∈ B , whose homotopy groups aregiven by π n Σ ∞ X = π Sn X = lim −→ k π n + k Σ k X . The relevant orthogonal spectrum is just (Σ ∞ X )( V ) = S V ∧ X .We will mention one important example of infinite complexes X and Y for which [Σ ∞ X, Σ ∞ Y ]is well-understood. Let G be a finite group, with classifying space BG . Let AG + be the set ofisomorphism classes of finite sets with a G -action. We can define addition and multiplication on AG + by [ X ] + [ Y ] = [ X ∐ Y ] and [ X ][ Y ] = [ X × Y ]. There are no additive inverses, but we canformally adjoin them to get a ring called AG , the Burnside ring of G ; this is not hard to workwith explicitly. There is a ring map ǫ : AG −→ Z defined by ǫ ([ X ] − [ Y ]) = | X | − | Y | , with kernel I say. We then have a completed ring b AG = lim ←− n AG/I n . The Segal conjecture (which was provedby Carlsson [10]) gives an isomorphism b AG ≃ [Σ ∞ BG + , Σ ∞ S ]. One can deduce a description of[Σ ∞ BG + , Σ ∞ BH + ] in similar terms for any finite group H .(b) For any virtual vector bundle V over any space X , there is a Thom spectrum X V ∈ B . In particular,if V is the tautological virtual bundle (of virtual dimension zero) over the classifying space BU ,then there is an associated Thom spectrum, normally denoted by M U . This has the property thatthe groups
M U ∗ X (as in Section 3) are given by π ∗ ( M U ∧ Σ ∞ X + ) (this is proved by a geometricargument, and is essentially the first step in the calculation of M U ∗ ). One can construct M U (andalso
M O and
M SO ) as strictly commutative ring spectra.Bott periodicity gives an equivalence BU ≃ Ω SU . A range of different proofs can be found in [5;28, Chapter 11; 41, Section 23; 45, Sections 6.4 and 8.8]. The filtration of Ω SU by the subspacesΩ SU ( k ) gives a filtration of M U by subspectra X ( m ), which are important in the proof of theHopkins-Devinatz-Smith nilpotence theorem. There are models of these homotopy types that arestrict ring spectra, but they cannot be made commutative.(c) For any generalized cohomology theory A ∗ , there is an essentially unique spectrum A ∈ B with A n X = [Σ ∞ X + , Σ n A ] for all spaces X and n ∈ Z . Similarly, for any generalized homology theory B ∗ , there is an essentially unique spectrum B ∈ B with B n X = π n ( B ∧ Σ ∞ X + ) for all spaces X and n ∈ Z . These facts are known as Brown representability ; the word “essentially” hides somesubtleties.(d) In particular, for any abelian group A there is an Eilenberg-MacLane spectrum HA ∈ B such that[Σ ∞ X + , Σ n HA ] = H n ( X ; A ) and π n ( HA ∧ Σ ∞ X + ) = H n ( X ; A ). (In fact, Brown representabilityis not needed for this: we have an explicit orthogonal spectram HA where HA ( V ) is the tensorproduct of A with the free abelian group generated by V , equipped with a suitable topology.) If A is a commutative ring, then HA is a strictly commutative ring spectrum. It is common to consider he case A = Z /
2. Here it can be shown that π ∗ (( H Z / ∧ ( H Z / Z / ξ , ξ , ξ , . . . ] , with | ξ k | = 2 k −
1. This is known as the dual Steenrod algebra , and denoted by A ∗ . The dual group A k = Hom( A k , Z /
2) can be identified with [ H Z / , Σ k H Z / Steenrod algebra . This ring is noncommutative, but its structurecan be described quite explicitly. It is important, because the mod 2 cohomology of any space (orspectrum) has a natural structure as an A ∗ -module. There is a similar story for mod p cohomologywhen p is an odd prime, but the details are a little more complicated.(e) Another consequence of Brown representability is that there is a spectrum I ∈ B such that [ X, I ] ≃ Hom( π X, Q / Z ) for all X ∈ B . This is called the Brown-Comenetz dual [7] of S ; it is geometricallymysterious, and a fertile source of counterexamples.(f) If M ∗ is a flat module over M U ∗ , then the functor X M ∗ ⊗ MU ∗ M U ∗ X is a homology theory,so there is a representing spectrum M with π ∗ ( M ∧ X ) = M ∗ ⊗ MU ∗ M U ∗ X . The Landweber exactfunctor theorem shows that flatness is not actually necessary: a weaker condition called Landweberexactness will suffice [30]. This condition is formulated in terms of formal group theory, and is ofteneasy to check in practice.Often M ∗ is a ring, and the M U ∗ -module structure arises from a ring map M U ∗ → M ∗ , whichcorresponds (by Quillen’s description of M U ∗ ) to a formal group law over M ∗ .Important examples include the Johnson-Wilson spectra E ( p, n ), with E ( p, n ) ∗ = Z ( p ) [ v , . . . , v n ][ v − n ](where | v k | = 2( p k − K ( p, n ) ∗ = Z /p [ v n , v − n ] is naturally a quotient E ( p, n ) ∗ /I n , where I n = ( p, v , . . . , v n − ).It it is not Landweber exact, but a corresponding spectrum K ( p, n ) can be constructed by othermeans. It is also useful to consider the completed spectra b E ( p, n ) with π ∗ ( b E ( p, n )) = ( E ( p, n ) ∗ ) ∧ I n = Z p [[ v , . . . , v n − ]][ v ± n ] . This is again Landweber exact.When M ∗ is a ring, one might hope to find a model of this homotopy type that is actually astrict ring spectrum, preferably strictly commutative. Unfortunately, this does not work very well.Often there will be uncountably many different ways to make M into a strict ring spectrum, withno way to pick out a preferred choice. Moreover, there will often not be any choice that is strictlycommutative. However, by a theorem of Hopkins and Miller, there is an essentially unique strictlycommutative model for b E ( p, n ). The reason why this case is special involves quite deep aspects ofthe algebraic theory of formal groups.If M ∗ is an algebra over M U ∗ that does not satisfy the Landweber criterion, one can still try toproduce a corresponding spectrum M by other methods. As mentioned previously, the paper [56]contains results in this direction.(g) For any small symmetric monoidal category category A , there is a K -theory spectrum K ( A ) ∈B . Computationally, this is very mysterious, apart from the fact that it is always connective (ie π n K ( A ) = 0 for n <
0) and π K ( A ) is the group completion of the monoid of connected componentsin A . Thomason has shown [58] that for every connective spectrum X there exists A with K ( A ) ≃ X . – If A is the category of finite sets and isomorphisms, then K ( A ) = Σ ∞ S . – Let G be a finite group, and let A be the category of finite G -sets and isomorphisms. Let A f (resp A t ) be the subcategory of free (resp. transitive) G -sets. Then K ( A ) = Σ ∞ B ( A t ) + , whichcan also be described as the wedge over the conjugacy classes of subgroups H ≤ G of the spectraΣ ∞ BW G H + , or as the fixed point spectrum of the G -equivariant sphere spectrum in the senseof Lewis-May-Steinberger [34]. On the other hand, K ( A f ) = Σ ∞ BG + . – Work of Kathryn Lesh [32] can be interpreted as exhibiting symmetric monoidal categories M n (of “finite multisets with multiplicities at most n ”) whose K -theory is the n ’th symmetric powerof Σ ∞ S . – If A is the symmetric monoidal category with object set N and only identity morphisms, then K ( A ) = H Z . One can set up a category A , whose objects are smooth compact closed 1-manifolds, andwhose morphisms are cobordisms between them. With the right choice of details, the K -theoryspectrum K ( A ) is then closely related to the classifying space of the stable mapping class group,and an important theorem of Madsen and Weiss [37] can be interpreted as saying that K ( A ) isthe Thom spectrum of the negative of the tautological bundle over C P ∞ , up to adjustment of π − . – If R is a commutative ring and A is the category of finitely generated projective R -modules,then K ( A ) is the algebraic K -theory spectrum usually denoted by K ( R ). Even in the case R = Z , this contains a great deal of arithmetic information. By rather different methodsone can construct spectra called T HH ( R ) and T C ( R ) (topological Hochschild homology andtopological cyclic homology) that approximate K ( R ); there is an extensive literature on theseapproximations. The definitions can be set up in such a way that the spectra K ( R ), T HH ( R )and T C ( R ) are all strictly commutative ring spectra.(h) The above construction can be modified slightly to take account of a topology on the morphism setsof A . We can then feed in the category of finite-dimensional complex vector spaces and isomorphisms(or a skeleton thereof) to get a spectrum known as kU , the connective complex K -theory spectrum,with a homotopy element u ∈ π kU such that π ∗ kU = Z [ u ]. This has the property that for finitecomplexes X , the group kU X = [Σ ∞ X + , kU ] is the group completion of the monoid of isomorphismclasses of complex vector bundles on X , or in other words the K -theory of X as defined by Atiyah andHirzebruch [2] (inspired by Grothendieck’s similar definition in the context of algebraic geometry).By a colimit construction, one can build a periodized version called KU with π ∗ KU = Z [ u, u − ].There are direct constructions of kU and KU using Bott periodicity rather than symmetric monoidalcategories. There are also constructions with a more analytic flavour, based on spaces of Fredholmoperators and so on [29]. With the right construction, both kU and KU are strictly commutativering spectra. It can be shown that KU is naturally a Landweber exact M U -algebra, so KU ∗ ( X ) = KU ∗ ⊗ MU ∗ M U ∗ ( X ) for all X .The infinite complex projective space C P ∞ is well-known to be a commutative group up tohomotopy. Using this, one can make the spectrum R = Σ ∞ ( C P ∞ ) + into a ring spectrum. Thestandard identification C P = S gives rise to an element v ∈ π ( R ), and we can use a homotopycolimit construction to invert this element, giving a new ring spectrum R [ v − ]. It is a theorem ofSnaith [55] that R [ v − ] is homotopy equivalent to KU .(i) Let F be a functor from based spaces to based spaces. Under mild conditions, we can use thehomeomorphism S ∧ S n −→ S n +1 to get a map S −→ Map( S n , S n +1 ) F −→ Map(
F S n , F S n +1 ) , and thus an adjoint map Σ F S n −→ F S n +1 . This gives a sequence of spectra Σ − n Σ ∞ F S n , whosehomotopy colimit (in a suitable sense) is denoted by D F . This is called the linearization or firstGoodwillie derivative of F . Goodwillie [19–21] has set up a “calculus of functors” in which thehigher derivatives are spectra D n F with an action of Σ n . (The slogan is that where the ordinarycalculus of functions has a denominator of n !, the calculus of functors will take coinvariants underan action of Σ n .) Even the derivatives of the identity functor are interesting; they fit in an intricateweb of relationships with partition complexes, symmetric powers of the sphere spectrum, Steinbergmodules, free Lie algebras and so on. There are other versions of calculus for functors from othercategories to spaces, with applications to embeddings of manifolds, for example.(j) A Moore spectrum is a spectrum X for which π n ( X ) = 0 when n < H n ( X ) = 0 when n > H be the category of Moore spectra. The functor H : H →
Ab is then close to being anequivalence: for any
X, Y ∈ H there is a natural short exact sequenceExt( H ( X ) , H ( Y )) / −→ [ X, Y ] −→ Hom( π ( X ) , π ( Y )) . Moreover, given any abelian group A there is a Moore spectrum SA (unique up to non-canoncalisomorphism) with H ( SA ) ≃ A .(k) Let C be an elliptic curve over a ring k . (Number theorists are often interested in the case where k isa small ring like Z , but it is also useful to consider larger rings like Z [ , c , c ][( c − c ) − ] that have arious universal properties in the theory of elliptic curves.) From this we obtain a formal group b C , which can be thought of as the part of C infinitesimally close to zero. It often happens thatthere is a spectrum E that corresponds to b C under the standard dictionary relating formal groupsto cohomology theories. Spectra arising in this way are called elliptic spectra [3]. The details areusually adjusted so that π ∗ ( E ) = k [ u, u − ] with k = π ( E ) and u ∈ π ( E ). In many cases E can beconstructed using the Landweber Exact Functor Theorem, as in (f).The spectrum T M F (standing for topological modular forms ) “wants to be” the universal exampleof an elliptic spectrum [14]. It is not in fact an elliptic spectrum, but it is close to being one, andit admits a canonical map to every elliptic spectrum. If we let
M F ∗ denote the group of integralmodular forms as defined by number theorists (graded so that forms of weight k appear in degree2 k ) then we have π ∗ ( T M F )[ ] = Z [ , c , c ][( c − c ) − ] = M F ∗ [ ] . The significance of the number 6 here is that 2 and 3 are the only primes that can divide | Aut( C ) | ,for any elliptic curve C . If we do not invert 6 then the homotopy groups π ∗ ( T M F ) are completelyknown, but different from
M F ∗ and too complex to describe here.There is a dense network of partially understood interactions between elliptic spectra, conformalnets [15], vertex operator algebras, chiral differential operators [11] and mathematical models ofstring theory [57]. It seems likely that some central aspects of this picture remain to be discovered.Under (g) we discussed the algebraic K -theory spectrum K ( R ) associated to to a commutativering R . The construction can be generalised to define K ( R ) when R is a commutative ring spectrum;in particular, we can define K ( kU ). Rognes and Ausoni [4] have found evidence of a relationshipbetween K ( kU ) and T M F , but many features of this remain obscure.(l) Let R be a strictly commutative ring spectrum, so for any space X we have a ring R ( X ) =[Σ ∞ X + , R ], and a group of units R ( X ) × . It can be shown that there is a spectrum gl ( R ) suchthat R ( X ) × = [Σ ∞ X + , gl ( R )] for all X . By taking X = S n we see that π n ( gl ( R )) = π n ( R ) for n >
0, but H ∗ ( gl ( R )) is not closely related to H ∗ ( R ), and many aspects of the topology of gl ( R )are mysterious, even in the case R = S . For any ( − T one can build a strictlycommutative ring spectrum of homotopy type Σ ∞ (Ω ∞ T ) + (where Ω ∞ T is the homotopy colimit ofthe spaces Ω n T ( R n )). Ring maps from this to R then correspond to maps T → gl ( R ) of spectra.(m) The notion of Bousfield localisation [6, 50] provides an important way to construct new spectra fromold. The simplest kinds of Bousfield localisations are the arithmetic ones: if X is a spectrum and p is a prime number then there are spectra and maps X [ p ] i ←− X j −→ X ( p ) such that i induces anisomorphism π ∗ ( X )[ p ] = π ∗ ( X ) ⊗ Z [ p ] → π ∗ ( X [ p ])and j induces an isomorphism π ∗ ( X ) ( p ) = π ∗ ( X ) ⊗ Z ( p ) → π ∗ ( X ( p ) ) . Similarly, there is a map X → X Q inducing π ∗ ( X ) ⊗ Q ≃ π ∗ ( X Q ). These properties characterise X [ p ], X ( p ) and X Q up to canonical homotopy equivalence. Along similar lines, there is a p -adiccompletion map X → X ∧ p . If each homotopy group π k ( X ) is finitely generated, then we just have π k ( X ∧ p ) = π k ( X ) ∧ p = π k ( X ) ⊗ Z p . In the infinitely generated case the picture is more complicated,but still well-understood.Next, for any spectrum E there is a functor L E : B → B and a natural map i : X → L E X characterised as follows: the induced map E ∗ i : E ∗ X → E ∗ L E X is an isomorphism, and if f : X → Y is such that E ∗ f is an isomorphism, then there is a unique map g : Y → L E X with gf = i . Thespectrum L E X is called the Bousfield localisation of X with respect to E , and we say that X is E -local if the map X → L E X is a homotopy equivalence. The slogan is that the category B E of E -local spectra is the part of stable homotopy theory that is visible to E . Apart from the arithmeticcompletions and localisations, the most important cases are the chromatic localisations L E ( p,n ) and L K ( p,n ) , which have been studied intensively [26, 50]. It is a key point here that E ( p, n ) and K ( p, n )have nontrivial homotopy groups in infinitely many negative degrees. If the homotopy groups of E nd X are bounded below, it turns out that L E X is just an arithmetic localisation of X , which isless interesting.(n) Under (b) we mentioned the spectra X ( n ) and M U = X ( ∞ ). If we localise at a prime p (as discussedin (m)) then these can be split into much smaller pieces. For example, we have π ∗ M U = Z [ x , x , . . . ]with | x i | = 2 i , but M U ( p ) is equivalent to a coproduct of suspended copies of a spectrum called BP ,where π ∗ BP = Z ( p ) [ v , v , . . . ] with | v k | = 2( p k − Brown-Peterson spectrum .There are also spectra called T ( n ) that have the same relation to X ( p n ) as BP does to M U . Thiswhole story is most naturally understood in terms of formal groups and p -typical curves. The spectra T ( n ) play an important role in a number of places, including the proof of the Nilpotence Theorem,the work of Mahowald, Ravenel and Shick on the Telescope Conjecture [38], and Ravenel’s methodsfor computation of stable homotopy groups of spheres [49].(o) The theory of surgery aims to understand compact smooth manifolds by cutting them into simplerpieces and reassembling the pieces [36]. A key ingredient is as follows: if we have an n -dimensionalmanifold with an embedded copy of S i − × B j (where i + j = n + 1), then we can remove the interiorto leave a manifold with boundary S i − × S j − , then glue on a copy of B i × S j − to obtain a newclosed manifold M ′ . If the original copy of S i − × B j is chosen appropriately, then the cohomologyof M ′ will be smaller than that of M . By iterating this process, we hope to convert M to S n (thenwe can reverse the steps to obtain a convenient description of M ). There are various obstructions tocompleting this process (and similar processes for related problems), and it turns out that these canbe encoded as problems in stable homotopy theory. Cobordism spectra such as M SO play a role, asdo the spectra kO and gl ( S ). When the dimension n is even and M is oriented, the multiplicationmap H n/ ( M ) ⊗ H n/ ( M ) −→ H n ( M ) = Z gives a bilinear form on H n/ ( M ), which is symmetricor antisymmetric depending on the parity of n/
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