I have a certain pointed path-connected topological space $X$, and a natural map $\mu : X \times X \to X$. I want to prove it is $H$-space. So I need to prove that maps $\mu(x_0,-)$ and $\mu(-,x_0)$ are homotopic to identity, where $x_0$ is base point of $X$.
In a reference I am reading, author says that proof follows from following fact: there is subspace $Y \subset X$ which is path-connected and contains base point $x_0$, and whose fundamental group, which is identified with commutator subgroup of $\pi_1(X)$, is trivial.
Of course, this implies that fundmanetal group of $X$ is abelian. But I don't understand why this implies $X$ is H-space. Is this clear from the given data, or maybe something specific of my space $X$?
Edit: It is obvious that not any arbitrary map $\mu$ will give an $H$-space. I guess the author means that the facts given make it somehow easier to check that $\mu$ is unital multiplication. Probably, they reduce it to some trivial verifications. But I am not sure why.