Homotopy groups of wedge sums of spectra

Question

This question came up when I was trying to understand Lemma 2.2.9 in Barnes & Rotzheim, which states that for any set of (sequential) spectra $X_i$, the natural map $$\bigoplus_i\pi_n(X_i)\to\pi_n\left(\bigvee_iX_i\right)$$ is an isomorphism.

If the $X_i$'s are CW-spectra (the case treated in this question), I can see that this follows from Hilton's theorem. I expect that the general case follows from this by some argument using CW approximation, but I cannot figure out the details, the main difficulty being that wedge sum does not preserve weak homotopy equivalences (counterexample).

However, since the homotopy groups of a spectrum can be alternatively defined as the colimit of the stable homotopy groups at each level, we only need the stable homotopy groups to be preserved. Is that true? If not, how can we prove the lemma for general spectra? (Barnes & Roitzheim assume that the spaces are CGWH, if that matters)

Answer 1

It is a great idea to use the fact that $\pi_*$ of a spectrum can be defined as a colimit of the stable homotopy groups of its spaces. As you point out, it suffices to show that for spaces $\pi_*^s(X \vee Y) \cong \pi^s_*(X) \oplus \pi_*^s(Y)$. This is true as a consequence of the Fruedenthal suspension theorem, which essentially states that "in a stable range" unstable homotopy groups behave like a homology theory, and this stable range grows with connectivity. When one forms the stable homotopy groups through a limiting process, the stable range goes to infinity and so the stable homotopy groups form a homology theory. In particular, they send wedge sums to direct sums.

An important corollary of this fact is that finite wedge sums are both a coproduct and product in the category of spectra.

Answer 2

In the homotopy category of spectra, the wedge is the coproduct operation (see for example, Adams' blue book, part III, discussion between Lemma 3.8 and Proposition 3.9; p. 172 in the linked version). Furthermore, the sphere object is small, meaning that mapping out of it commutes with coproducts — this is essentially because the sphere is compact as a topological space. Therefore we get $$ \pi_n\left(\bigvee X_i\right) = [S^n, \bigvee X_i] \cong \bigoplus [S^n, X_i] = \bigoplus \pi_n (X_i). $$ See also Margolis, Spectra and the Steenrod Algebra, chapter 2, where he verifies his axioms for spectra: existence of coproducts (p. 28) and smallness of the sphere object (p. 31).

Answer 3

Thanks to the other answerers for their insightful responses! I was trying to avoid assumptions that the levels of the sequential spectra are CW complexes (or well-pointed), since this is the setting in Barnes & Roitzheim (though I imagine it may end up not mattering in the homotopy category). It seems like that Margolis' arguments and the Freudenthal suspension theorem both have assumptions of this form. Another argument I found in Hatcher (Proposition 4F.1) also depends essentially on a CW structure.

After some more searching, I found Schwede's Symmetric Spectra, whose Proposition 2.19(i) addressed this exact problem. Here is an outline of the argument:

Let $X$ and $Y$ be two sequential spectra. Start with the long exact sequence (which only exists stably) of homotopy groups for the homotopy cofiber sequence of spectra $X\stackrel{i_X}{\to} X\vee Y\to Ci_X\simeq CX\vee Y\simeq Y$ (where $\simeq$ denotes levelwise homotopy equivalence): $$\cdots\to\pi_n(X)\to\pi_n(X\vee Y)\stackrel\alpha\to \pi_n(Y)\to\pi_{n-1}(X)\to\cdots$$ Since $i_X$ admits a retract, $\pi_n(i_X)$ is injective. Since $\alpha$ is induced by the collapse map $X\vee Y\to Y$, which admits a section, we have $$\pi_n(X\vee Y)\cong \pi_n(X)\oplus\pi_n(Y).$$ The general case follows from a compactness argument and induction.

I believe this doesn't have any assumption on the spaces, please correct me if I'm wrong.