Stable Homotopy theory


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The notion of CW-spectrum is the analogue for sequential spectra in Top of the concept of CW-complex for topological spaces.

Just like CW-complexes are cofibrant objects in the classical model structure on topological spaces, so CW-spectra are cofibrant objects in the stable model structure on topological sequential spectra, see below.


In components


A CW-spectrum X SeqSpec(Top)X_\bullet\in SeqSpec(Top) is a sequential spectrum in Top such that

  1. all component spaces X nX_n are CW-complexes,

  2. all structure maps ΣX nX n+1\Sigma X_n \longrightarrow X_{n+1} are inclusions of subcomplexes

e.g. (Adams 74, p. 139) Beware that for instance (Switzer 75, def. 8.1) says just “spectrum” for “CW-spectrum”.

For a CW-spectrum XX there is a concept of “cell of a spectrum”:


A cell of a CW-spectrum, def. is a cell of one of the components CW-complexes X nX_n, together with all its suspensions in all the higher component spaces X >nX_{\gt n}, subject to the condition that the first cell itself is not itself the suspension of a cell in X n1X_{n-1}.

This way every CW-spectrum is the union of all its cells in the sense of def. . (e.g. Switzer 75, 8.4).


A CW-spectrum, def. , is called a finite spectrum (or countable spectrum, etc.) if it has finitely many cells (countably many cells) according to def. .

Via spectrum attaching maps


For nn \in \mathbb{Z} (possibly negative) define 𝕊 n\mathbb{S}^n to be the sequential prespectrum with component spaces

(𝕊 n) k{S k+n ifk+n0 * otherwise (\mathbb{S}^{n})_k \coloneqq \left\{ \array{ S^{k+n} & if \; k + n \geq 0 \\ \ast & otherwise } \right.

and with structure maps the canonical isomorphisms.

(Lewis-May-Steinberger 86, def. 4.3)


In def.

  • for n=0n = 0 then 𝕊 0=𝕊=Σ S 0\mathbb{S}^0 = \mathbb{S} = \Sigma^\infty S^0 is standard sequential incarnation of the sphere spectrum;

  • for nnn \geq n then 𝕊 nΣ S n\mathbb{S}^n \simeq \Sigma^\infty S^n is the suspension spectrum on the n-sphere;

  • for general nn then 𝕊 nF nS 0\mathbb{S}^n \simeq F_{-n} S^0 is also known as the (n)(-n)th free spectrum on S 0S^0.


A cell spectrum is a topological sequential spectrum XX realized as the colimit over a sequence of spectra *=X 0X 1X 2X 3\ast = X_0 \to X_1 \to X_2 \to X_3 \to \cdots such that there are morphisms

j n:(iI n𝕊 q n)X n j_n \;\colon\; \left( \underset{i \in I_n}{\sqcup} \mathbb{S}^{q_n} \right) \longrightarrow X_n

with X n+1=Cone(j n)X_{n+1}= Cone(j_n) (the mapping cone).


A cell spectrum is a CW-spectrum if each attaching map Σ S q nX n\Sigma^\infty S^{q_n}\to X_n factors through a X kX nX_k \to X_n with k<qk \lt q.

(e.g. Lewis-May-Steinberger 86, def. 5.1, def. 5.2, Weiss)

Category of CW-spectra

There is an obvious category of CW-spectra given as the full subcategory CWSpecCWSpec' of SeqSpec(Top)SeqSpec(Top) consisting of the CW-spectra, ie. a morphism of CW-spectra f:XYf: X \to Y is given by levelwise maps f n:X nF nf_n: X_n \to F_n that are compatible with the structure maps. In accordance with Switzer 75, 8.9, we call morphisms in this sense functions.

However, often there is a more useful notion of morphisms between CW-spectra.


Let EE be a CW-spectrum. A subspectrum is a CW spectrum FF such that each F nF_n is a subcomplex of E nE_n. A subspectrum is cofinal if for each cell e nE ne_n \in E_n, there is some mm such that Σ me nF n+m\Sigma^m e_n \in F_{n + m}.

We can then construct the category CWSpecCWSpec as the localization of CWSpecCWSpec' at the cofinal inclusions. Since the class of cofinal inclusions admit a calculus of right fractions, the cateogry CWSpecCWSpec has the following concrete description - a morphism XYX \to Y in CWSpecCWSpec is given by a function X˜Y\tilde{X} \to Y, where X˜\tilde{X} is some cofinal subspectrum of XX, quotiented by the relation that two such morphisms f:X˜Yf: \tilde{X} \to Y and f:X˜Yf': \tilde{X}' \to Y are considered equivalent if there is some further cofinal subspectrum X˜X˜X˜\tilde{X}'' \subseteq \tilde{X} \cap \tilde{X'} such that f| X˜=f| X˜f|_{\tilde{X}'} = f'|_{\tilde{X}''}.




A sequential spectrum XSeqSpec(Top) stableX\in SeqSpec(Top)_{stable} is cofibrant in the stable model structure on topological sequential spectra in particular if all component spaces are cell complexes and all its structure morphisms S 1X nX n+1S^1 \wedge X_n \to X_{n+1} are relative cell complexes. In particular CW-spectra, def. , are cofibrant in SeqSpec(Top) stableSeqSpec(Top)_{stable}.

For the proof see there.


For XSeqSpec(Top) stableX\in SeqSpec(Top)_{stable} a CW-spectrum, then its standard cylinder spectrum X(I +)X \wedge (I_+) is a good cylinder object, in that the inclusion

XXX(I +) X \vee X \longrightarrow X \wedge (I_+)

is a cofibration in SeqSpec(Top) stableSeqSpec(Top)_{stable}.

See this prop..


The analog of CW-approximation for topological spaces holds true for topological sequential spectra:


For XSeqSpec(Top)X \in SeqSpec(Top) a topological sequential spectrum, there exists a CW-spectrum X^\hat X and a stable weak homotopy equivalence

X^X. \hat X \longrightarrow X \,.

(e.g. Elmendorf-Kriz-May 95, theorem 1.5)

Proof 1

First let X^ 0X 0\hat X_0 \longrightarrow X_0 be a CW-approximation of the component space in degree 0, via this prop.. Then proceed by induction: suppose that for nn \in \mathbb{N} a CW-approximation ϕ kn:X^ knX kn\phi_{k \leq n} \colon \hat X_{k \leq n} \to X_{k \leq n} has been found such that all the structure maps are respected. Consider then the continuous function

ΣX^ nΣϕ nΣX nσ nX n+1. \Sigma \hat X_n \overset{\Sigma \phi_n}{\longrightarrow} \Sigma X_n \overset{\sigma_n}{\longrightarrow} X_{n+1} \,.

Applying that prop. to this function factors it as

ΣX nX^ n+1ϕ n+1X n+1. \Sigma X_n \hookrightarrow \hat X_{n+1} \overset{\phi_{n+1}}{\longrightarrow} X_{n+1} \,.

Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property:

ΣX^ n Σϕ n ΣX n incl σ n X^ n+1 ϕ n+1 X n+1. \array{ \Sigma \hat X_n &\overset{\Sigma \phi_n}{\longrightarrow}& \Sigma X_n \\ {}^{incl}\downarrow && \downarrow^{\mathrlap{\sigma_n}} \\ \hat X_{n+1} &\underset{\phi_{n+1}}{\longrightarrow}& X_{n+1} } \,.
Proof 2

A high-powered way to see this is to use the Quillen equivalence between the stable model structure on topological sequential spectra and the stable Bousfield-Friedlander model structure (see there) on sequential spectra in simplicial sets. This implies that a CW-approximation is given by

|QSingX|W stableX, {\vert Q Sing X\vert} \overset{\in W_{stable}}{\longrightarrow} X \,,

where ||Sing{\vert - \vert} \dashv Sing is degreewise the adjunction between geometric realization and forming singular simplicial complex, and QQ denotes any cofibrant replacement in the BF-model structure.


Discussion in the generality of equivariant spectra is in

  • L. Gaunce Lewis, Peter May, and M. Steinberger (with contributions by J.E. McClure), section I.5 of Equivariant stable homotopy theory Springer Lecture Notes in Mathematics Vol.1213. 1986 (pdf)

Last revised on September 5, 2016 at 09:17:19. See the history of this page for a list of all contributions to it.