An H-space $(X,\mu,e)$, is a pointed topological space $(X,e)$ together with a continuous map $\mu:X\times X\to X$, $s.t.$, The maps $\mu(\cdot,e):X\to X$ and $\mu(e,\cdot):X\to X$ are both homotopy to the identity $id:X\to X$ relative to $\{e\}$.
In Hatcher's Exercise 3.C.1(p.291), he claim that For a CW complex, the definition can be weaken:
let $e \in X$ be a 0 -cell, then $X$ is an H-space if there is a map $\mu: X \times X \rightarrow X$ such that $\mu(\cdot, e)$ and $\mu(e, \cdot)$, are homotopic to the identity(NOT necessarily rel$\{e\}$).
Or strengthen
With the same hypothesis, $\mu$ can be homotoped so that $e$ is a strict identity.
I was looking for a solution and I found this. However, there is some mistake in his proof. The existence of his assembled homotopy is equivalent to that $X$ is an H-space. He used the result that he was going to prove.
Although he didn't make it clear, the idea is worth taken, that is, deduce some homotopy $(X\vee X)\times I\to X$ and extend it to $X\times X$ by HEP. I didn't see how to do this, can anyone give some help?
