H-space

Remarks

Some authors drop the condition that the homotopies fix $e$; for CW complexes the two definitions give the same class of spaces. An H-space is homotopy associative if $\mu \circ (\mu \times \id) \simeq \mu \circ (\id \times \mu)$, and homotopy commutative if $\mu \simeq \mu \circ \tau$, where $\tau$ swaps the factors.

Example: every topological group $G$ with $e$ its identity and $\mu$ its multiplication is an H-space, since $\mu(e, x) = \mu(x, e) = x$ on the nose.

Example: the loop space $\Omega Y$ with concatenation of loops is an H-space; the constant loop is a unit up to the standard reparametrization homotopy.

Non-example: $S^2$ admits no such $\mu$. If it did, let $x \in H^2(S^2; \Z)$ be a generator; the unit condition forces $\mu^*(x) = x \times 1 + 1 \times x$ in $H^2(S^2 \times S^2; \Z)$. Then $0 = \mu^*(x^2) = (x \times 1 + 1 \times x)^2 = 2 (x \times x)$ for the cup product, which is false since $x \times x$ generates $H^4(S^2 \times S^2; \Z) \cong \Z$.