nLab ordinary homology spectra split

Statement

For $S$ any spectrum and $H A$ an Eilenberg-MacLane spectrum, then the smash product $S\wedge H A$ (the $A$-ordinary homology spectrum) is non-canonically equivalent to a product of EM-spectra (hence a wedge sum of EM-spectra in the finite case).

(Adams 74, part II, lemma 6.1)

A variant for generalized (Eilenberg-Steenrod) cohomology:

Let $X$ be a topological space such that each of the ordinary homology groups $H_n(X,\mathbb{Z})$ is a free abelian group on genrators $\{h_{\alpha,n}\}_{\alpha \in B_n}$. Write $c_{\alpha,n} \in H^n(X,\mathbb{Z}) \simeq Hom(H_n(X,\mathbb{Z}), \mathbb{Z})$ for the corresponding dual basis.

Let $E$ be a multiplicative cohomology theory and write $h'_{n,\alpha} \in (\tau_{\leq 0} E)_n(X)$ and $c'_{n,\aloha} \in (\tau_{\leq 0} E)^n(X)$for the images of these generators under the 0-truncated unit map

$$H \mathbb{Z} \simeq \tau_{\leq 0} \mathbb{S} \stackrel{}{\longrightarrow} \tau_{\leq 0} E \,.$$

Proposition

If one of the following conditions is satisfied

• Each $h'_{n,\alpha}$ lifts through $E_n(X) \to (\tau_{\leq 0}E)_n(X)$;

• each $H_n(X,\mathbb{Z})$ is finitely generated and each $c'_{n,\alpha}$ lifts through $E^n(X) \to (\tau_{\leq 0}E)^n(X)$,

then there are non-canonical equivalences as follows:

1. $E \wedge \Sigma^\infty X_+ \simeq \underset{n,\alpha}{\vee} \Sigma^n E \;\; \in E Mod$ ;

2. $[X,E] \simeq \underset{n,\alpha}{\prod} \Sigma^{-n} E$;

3. $E_\bullet(X) \simeq \pi_\bullet(E) \otimes H_\bullet(X,\mathbb{Z})$

   and

   $E^\bullet(X)\simeq Hom(H_\bullet(X), \pi_\bullet(E))$

(Lurie 10, lecture 4, prop. 7, Adams 74, part II, lemma 2.5)

Related concepts

• splitting principle

References

• Frank Adams, Stable homotopy and generalised homology, 1974

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 (web), Lecture 4 Complex-oriented cohomology theories (pdf)