Let $H$ be an $H-$space with a multiplication $H \times H \xrightarrow{{ \circ}} H$. Someone told me that a multiplication on an H-space is a map $H \wedge H \xrightarrow{} H$ satisfying some properties. The implication is that the map $H \times H \to H$ factors through $H \wedge H$(up to homotopy). I don't agree with this:
Suppose it did. Let $pt$ be the homotopy identity in $H$. Then $H \times {pt} \hookrightarrow H \times H \to H$ would be homotopic to $H \times pt \to H \times pt \sqcup_{pt \times pt} pt \times H \hookrightarrow H \times H \to H \wedge H \to H$ which is the null map. But the above map is homotopic to the identity map. So this description of a multiplication doesn't make sense. Help!
The problem is that we are taking the smash product with the identity element as the basepoint.
An $H$-space structure on $H$ is still a pointed map $H_+ \wedge H_+ \to H_+$