THE SPHERE SPECTRUM

John Rognes

October 28th 2004

Following J.H.C. Whitehead [14] and Lima [7], a sequential spectrum E is a sequence of based topological spaces (or simplicial sets) E_{n} and structure maps σ_{n}: ΣE_{n} → E_{n+1}, for all n ≥ 0. Here the suspension ΣE_{n} = S^{1} ∧ E_{n} equals the smash product with the based topological (or simplicial) circle S^{1}. The sphere spectrum 𝕊 is the most basic example: its n-th space is the n-sphere S^{n} = S^{1} ∧⋯∧ S^{1} and its structure maps are the resulting homeomorphisms σ_{n}: ΣS^{n} → S^{n+1}.

The k-th homotopy group π_{k}(E) is the colimit over n of the (unstable) homotopy groups π_{k+n}(E_{n}). As k varies, the π_{k}(E) assemble to a graded abelian group π_{∗}(E). In the case of the sphere spectrum, π_{k}(𝕊) is the colimit of the homotopy groups of spheres π_{k+n}(S^{n}), i.e., of the homotopy classes of based maps S^{k+n} → S^{n}. By the Freudenthal suspension theorem [5] the homomorphisms in this colimit are isomorphisms for n ≥ k + 2, and the common limiting value π_{k}(𝕊) is known as the k-th stable homotopy group of spheres.

The generalized homology theory π_{∗}^{S}(X) = π_{∗}(𝕊 ∧ X) and the generalized cohomology theory π_{S}^{−∗}(X) = π_{∗} Map(X, 𝕊) associated to the sphere spectrum are called stable homotopy and stable cohomotopy, respectively. As a consequence of the proven Segal conjecture [3], stable cohomotopy has the exceptional property that π_{S}^{−∗}(BG_{+}) vanishes for all ∗ < 0. Here BG is the classifying space of an arbitrary finite group. In the Atiyah–Hirzebruch spectral sequence

E_{s,t}^{2} = H^{−s}(BG; π_{t}(𝕊)) ⟹ π_{S}^{−(s+t)}(BG_{+})

there is a complicated differential interplay between group cohomology and the stable homotopy groups of spheres, making E_{s,t}^{∞} = 0 for s + t < 0. For G = ℤ/p the Segal conjecture also provides a copy of π_{∗}(𝕊)_{p}^{∧} as a direct summand in the abutment, so each class x ∈ π_{k}(𝕊)_{p}^{∧} is represented at E_{s,t}^{∞} by some coset M(x) ⊂ π_{t}(𝕊)_{p}^{∧}, with t ≥ k, called the Mahowald root invariant of x. Empirically, when x is part of a periodic family in π_{∗}(𝕊)_{p}^{∧} detected by the n-th Morava K-theory K(n), then M(x) is part of a family detected by the next Morava K-theory K(n + 1) [9].

A map of sequential spectra f: E → F is a sequence of based maps f_{n}: E_{n} → F_{n} commuting with the structure maps. It induces a homomorphism f_{∗}: π_{∗}(E) → π_{∗}(F) of homotopy groups, and is called a stable equivalence if f_{∗} is an isomorphism in each degree. The stable homotopy category is the category obtained from

Typeset by AMS-TEX

the category of spectra by inverting the stable equivalences. Let Sp denote the category of sequential spectra, and let B (for Boardman) denote its associated stable homotopy category. The sphere spectrum 𝕊 generates B in the sense that for each spectrum E there exists a cell spectrum E^{c}, which has been assembled from integer suspensions of 𝕊 in the same way that a cell complex is built from non-negative suspensions of S^{0}, and a stable equivalence E^{c} → E.

The smash product of two maps S^{k+m} → S^{m} and S^{ℓ+n} → E_{n} composes with the iterated structure map S^{m} ∧ E_{n} → E_{m+n} to produce a map S^{k+m+ℓ+n} → E_{m+n}. With some care, especially about the ordering of the various circle factors in these smash products, this rule induces a pairing π_{k}(𝕊) ⊗ π_{ℓ}(E) → π_{k+ℓ}(E). In the case E = 𝕊 this product makes π_{∗}(𝕊) a graded commutative ring, and in general π_{∗}(E) is a graded module over π_{∗}(𝕊).

In the stable homotopy category B there is a functorial smash product E ∧ F of spectra [1], well-defined up to stable equivalence, so that these commutative ring and module structures are realized by morphisms 𝕊 ∧ 𝕊 → 𝕊 and 𝕊 ∧ E → E. However, in the category Sp of sequential spectra there is no definition of a smash product E ∧ F such that the product on 𝕊 is commutative. It is at best associative, and sequential spectra E and F may be regarded as left (or right) 𝕊-modules, but no natural 𝕊-module structure remains on their smash product E∧F. The situation is reminiscent of that of modules over a non-commutative ring.

To overcome this defect, modern stable homotopy theory takes place in one of several possible modified categories Sp′ of spectra, three of which are reviewed below. In each of these there is a smash product E∧F defined within the category of spectra, that is so well-behaved that the sphere spectrum 𝕊 admits a commutative product 𝕊 ∧ 𝕊 → 𝕊 in Sp′. More precisely, the smash product is a symmetric monoidal pairing (= coherently unital, associative and commutative) with 𝕊 as the unit object. The spectra E, F are naturally modules over 𝕊 with this product, i.e., 𝕊-modules, and the smash product E ∧ F over 𝕊 of two 𝕊-modules is again an 𝕊-module, because 𝕊 is commutative. Furthermore, there is a notion of stable equivalence on Sp′, so chosen that the associated homotopy category is equivalent to B.

This makes the sphere spectrum 𝕊 the initial ground “ring” for stable homotopy theory, much like the integers ℤ is the initial ground ring for algebra. The categories of 𝕊-modules, resp. associative or commutative 𝕊-algebras, can be thought of as enriched versions of the categories of ℤ-modules (= abelian groups), resp. associative or commutative ℤ-algebras (= rings). This is a fruitful point of view for promoting ideas from algebra, algebraic geometry or number theory to the algebraic-topological context. The Eilenberg–Mac Lane functor embeds algebra into topology, and the enrichment amounts to a change of ground ring along the Hurewicz map h: 𝕊 → ℤ. The earlier theories of A_{∞} and E_{∞} ring spectra [11] provide many more examples of associative and commutative 𝕊-algebras in topology, beyond those coming from algebra. These are therefore “brave new rings,” a term coined by Waldhausen.

Several modern reinterpretations Sp′ of the category of spectra appeared shortly after 1994. The principal three are (a) the S-modules M_{S} of Elmendorf, Kriz, Mandell and May [4], (b) the symmetric spectra Sp^{Σ} of Hovey, Shipley and Smith [6], and (c) the Γ-spaces ΓS_{∗} of Segal [13] and Lydakis [8]. The essential equivalence of these and other approaches is discussed in [10] and [12].

(a) The S-modules of May et al. were introduced in [4]. To start, a coordinatefree spectrum E is a rule that assigns a based space EV to each finite-dimensional vector subspace V ⊂ ℝ^{∞}, together with a compatible system of homeomorphisms EV ≅ Ω^{W −V} EW whenever V ⊂ W. Here W − V is the orthogonal complement of V in W, S^{W −V} is its one-point compactification, and Ω^{W −V} X = F (S^{W −V} , X) is the mapping space. The coordinate-free sphere spectrum S is the rule with SV = colim_{V ⊂W} Ω^{W −V} S^{W}. Its 0th space S0 is also known as Q(S^{0}).

An 𝕃-spectrum is a coordinate-free spectrum equipped with a suitable action by the space L(1) of linear isometries ℝ^{∞} → ℝ^{∞}, which is part of the linear isometries operad L. The sphere spectrum S is canonically an 𝕃-spectrum, and there is an operadic smash product E ∧_{L} F of 𝕃-spectra. Finally, the S-modules are the 𝕃spectra E such that a natural map λ: S ∧_{L} E → E is an isomorphism. The sphere spectrum is an S-module, and the operadic smash product of 𝕃-spectra E ∧_{L} F restricts to the desired smash product E ∧ F on the full subcategory of S-modules.

(b) The symmetric spectra of J. Smith et al. were introduced in [6]. First, a symmetric sequence E is a sequence of based simplicial sets E_{n} with an action by the symmetric group Σ_{n}, for each n ≥ 0. The sphere symmetric sequence 𝕊 has the n-fold smash product S^{n} = S^{1} ∧ ⋯ ∧ S^{1} as its n-th space, with Σ_{n} permuting the factors. There is a symmetric monoidal pairing E ⊗ F of symmetric sequences, so defined that a map E ⊗ F → G corresponds to a set of (Σ_{m} × Σ_{n})-equivariant maps E_{m} ∧ F_{n} → G_{m+n}. Then 𝕊 is a commutative monoid with product 𝕊 ⊗ 𝕊 → 𝕊 corresponding to the equivariant isomorphisms S^{m} ∧ S^{n} ≅ S^{m+n}.

A symmetric spectrum E is defined to be an 𝕊-module in symmetric sequences, i.e., a symmetric sequence with a unital and associative action 𝕊 ⊗ E → E. Explicitly, the module action amounts to a set of (Σ_{m} × Σ_{n})-equivariant maps S^{m} ∧ E_{n} → E_{m+n}. The sphere spectrum 𝕊 is then a symmetric spectrum, and the desired smash product E ∧F of two symmetric spectra is defined as the coequalizer of two obvious maps E ⊗ 𝕊 ⊗ F → E ⊗ F. There is a notion of a stable equivalence f: E → F of symmetric spectra, strictly more restrictive than asking that π_{∗}(f) is an isomorphism, so that the associated homotopy category is equivalent to B.

A variant of symmetric spectra, called orthogonal spectra [10], is obtained by replacing the symmetric group actions by orthogonal group actions. Then the π_{∗}-isomorphisms are the correct weak equivalences to invert, in order to obtain a homotopy category equivalent to B.

(c) Let Γ be the category of finite sets n_{+} = {0, 1, …, n} based at 0, for n ≥ 0, and base-point preserving functions. Segal [13] defined a Γ-space E to be a functor from Γ to based simplicial sets, such that E(0_{+}) is a point. Each Γ-space can be prolonged (degreewise) to an endofunctor of based simplicial sets, and there is an associated sequential spectrum with n-th space E(S^{n}). Bousfield and Friedlander [2] show that the homotopy category of Γ-spaces under stable equivalences is equivalent to the stable homotopy category of connective spectra, i.e., spectra with π_{k}(E) = 0 for k < 0.

The sphere Γ-space 𝕊 is the functor that takes n_{+} to itself, considered as a based simplicial set. Its prolongation is the identity endofunctor, and the associated sequential spectrum is the sphere spectrum 𝕊. The smash product E ∧ F of two Γ-spaces is defined so that a map E ∧ F → G of Γ-spaces amounts to a natural transformation E(k_{+}) ∧ F (ℓ_{+}) → G(k_{+} ∧ ℓ_{+}), for k_{+} and ℓ_{+} in Γ. This defines a symmetric monoidal pairing on Γ-spaces, with the sphere as the unit object. Lydakis [8] realized that this categorical construction also has good homotopical properties, in particular that it really models the smash product of spectra.

References

[1] Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill. – London, 1974.

[2] Bousfield, A. K.; Friedlander, E. M., Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130.

[3] Carlsson, Gunnar, Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. (2) 120 (1984), 189–224.

[4] Elmendorf, A. D.; Kriz, I.; Mandell, M. A.; May, J. P., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997.

[5] Freudenthal, Hans, Über die Klassen der Sphärenabbildungen. I. Große Dimensionen, Compos. Math. 5 (1937), 299–314.

[6] Hovey, Mark; Shipley, Brooke; Smith, Jeff, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.

[7] Lima, Elon L., Stable Postnikov invariants and their duals, Summa Brasil. Math. 4 (1960), 193–251.

[8] Lydakis, Manos, Smash products and Γ-spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999), 311–328.

[9] Mahowald, Mark E.; Ravenel, Douglas C., The root invariant in homotopy theory, Topology 32 (1993), 865–898.

[10] Mandell, M. A.; May, J. P.; Schwede, S.; Shipley, B., Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), 441–512.

[11] May, J. Peter, E_{∞} ring spaces and E_{∞} ring spectra. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave, Lecture Notes in Mathematics, vol. 577, Springer-Verlag, Berlin – New York, 1977.

[12] Schwede, Stefan, S-modules and symmetric spectra, Math. Ann. 319 (2001), 517–532.

[13] Segal, Graeme, Categories and cohomology theories, Topology 13 (1974), 293–312.

[14] Spanier, E. H.; Whitehead, J. H. C., A first approximation to homotopy theory, Proc. Nat. Acad. Sci. U. S. A. 39 (1953), 655–660.

Department of Mathematics, University of Oslo, Norway

   E-mail address: rognes@math.uio.no