# Wedge axiom of a homology theory like functor

Let $$h_n\colon CW_*\to Ab (n\in \mathbb{Z})$$ be a covariant homotopy invariant functor that sends cofibre sequences to exact sequences and is equipped with natural suspension isomorphisms.

Then the following two statements are equivalent:

1. For every countable collection $$(X_i)_{i\in I}\subseteq CW_*$$ the natural morphism $$\bigoplus_i h_*(X_i)\to h_*(\bigvee_i X_i)$$ is an isomorphism.

2. If $$Y=colim_k(Y_0\overset{cl.incl.}{\hookrightarrow}Y_1\hookrightarrow\cdots)$$, then the natural morphism $$colim_k h_*(Y_k)\to h_*(Y)$$ is an Isomorphism.

I tried to show this, and had some ideas, but I still lack of understanding for a detailed proof. In particular, it is not clear to me how the suspension isomorphisms and the cofibre condition have to be used.

My considerations are as follows:

$$2.\implies 1.$$ Assume 2. is true. Let $$(X_i)_i$$ be any countable collection of pointed CW-complexes. We may assume, that the index set are the natural numbers. The most obvious thing that comes to my mind is to let $$Y_k:=X_0\vee \cdots \vee X_k$$. Then let $$Y=colim(Y_0\hookrightarrow Y_1\hookrightarrow \cdots)$$.

(one issue here is, that I don't know how to replace the inclusions by closed ones, so I neglect this for a moment)

Now by 2. we know that the natural morphism $$colim_k h_*(Y_k)\to h_*(Y)$$ is an isomorphism.

My intuition tells me that $$h_*(X_0\vee\cdots \vee X_k)\cong h_*(X_0)\oplus \cdots \oplus h_*(X_k)$$ should be true under the above assumptions, and that one has $$colim_k (h_*(X_0)\oplus\cdots\oplus h*(X_k))\cong \bigoplus_i h_*(X_i)$$.

$$1.\implies 2.$$ Assume 1. is true. Let $$Y=colim(Y_0\hookrightarrow Y_1\hookrightarrow\cdots)$$, where the $$hookrightarrow$$ are closed inclusions. If one could make the $$Y_k$$ become disjoint, the relations between them would disappear and the colimit would become a coproduct. Hence one could replace $$Y$$ by the reduced mapping telescope $$\bigcup_{k\geq 0} [k,k+1]\times Y_k$$ and show that this doesn't affect the value of $$h_*$$. By one we would have $$\bigoplus_k h_*(Y_k)\cong colim_k h_*([k,k+1]\times Y_k)\cong colim_k h_*(Y_k)$$.

-Is there a way to make this argument work?

-How do the suspension isomorphisms and the cofibre condition join the argument?

• You consider a reduced homology theory on $CW_*$. It is well-known that then $h_*(X \vee Y) \approx h_*(X) \oplus h_*(Y)$ which generalizes to finite wedges. Moreover, the inclusion $X_1 \vee \dots \vee X_k \hookrightarrow X_1 \vee \dots \vee X_{k+1}$ is closed. – Paul Frost Jan 7 at 9:29
• 1. $\Rightarrow$ 2. is not all trivial. You can find a proof in Switzer, Robert M. Algebraic Topology - Homotopy and Homology. Springer, 2017. See Proposition 7.53 which has a two-pages-proof (plus the material proved previously). – Paul Frost Jan 7 at 13:28