Suppose that \(X\) is a CW complex with basepoint \(e \in X\) a 0-cell. Show that \(X\) is an H-space if there is a map \(\mu: X \times X \to X\) such that the maps \(X \to X\), \(x \mapsto \mu(x, e)\) and \(x \mapsto \mu(e, x)\), are homotopic to the identity. [Sometimes this is taken as the definition of an H-space, rather than the more restrictive condition in the definition we have given.] With the same hypotheses, show also that \(\mu\) can be homotoped so that \(e\) is a strict two-sided identity.
Solution
Because \(e\) is a \(0\)-cell of the CW complex \(X\), the inclusion \(\{e\}\hookrightarrow X\) is a closed cofibration; equivalently, \((X,e)\) is well-pointed. For well-pointed H-spaces there is a standard unit-strictification theorem: a multiplication having a two-sided homotopy unit in the unbased sense can be homotoped first so that the unit homotopies preserve the basepoint, and if \(X\vee X\hookrightarrow X\times X\) is a cofibration, the multiplication can then be homotoped to have a strict two-sided unit. See tom Dieck–Kamps–Puppe, Homotopy Theory, Theorem 3.37 and Proposition 3.38.
The hypotheses of that theorem hold here. Indeed \(\{e\}\hookrightarrow X\) is a closed cofibration, hence so is \[ X\vee X=(X\times\{e\})\cup(\{e\}\times X) \hookrightarrow X\times X. \] By assumption, the two restrictions \[ r(x)=\mu(x,e), \qquad \ell(x)=\mu(e,x) \] are homotopic to \(\operatorname{id}_X\). The first part of the strictification theorem therefore replaces \(\mu\), through a homotopy of multiplications, by a multiplication for which these homotopies can be taken through maps \((X,e)\to(X,e)\). This is exactly Hatcher’s H-space condition.
For the stronger assertion, apply the second part of the theorem to the cofibration \(X\vee X\hookrightarrow X\times X\). The restriction of the adjusted multiplication to \(X\vee X\) is homotopic, through based maps, to the fold map \[ \nabla:X\vee X\to X, \qquad \nabla|_{X\times\{e\}}=\operatorname{id} =\nabla|_{\{e\}\times X}. \] The homotopy extension property extends this homotopy from the wedge to all of \(X\times X\). At the end we obtain a multiplication \(\mu'\) homotopic to \(\mu\) such that \[ \mu'(x,e)=x=\mu'(e,x) \] for every \(x\in X\). Thus \(e\) is a strict two-sided identity.