Show that a retract of an H-space is an H-space if it contains the identity element.
Solution
Let \(X\) be an H-space with identity \(e\), and let \(A\subset X\) be a retract containing \(e\). Write \[ i:A\hookrightarrow X, \qquad r:X\to A, \qquad r\circ i=\operatorname{id}_A. \] If \[ \mu:X\times X\to X \] is the H-space multiplication, define \[ \mu_A=r\circ\mu\circ(i\times i):A\times A\to A. \] Then \[ \mu_A(a,e)=r(\mu(i(a),e)), \qquad \mu_A(e,a)=r(\mu(e,i(a))). \] The two maps \(x\mapsto\mu(x,e)\) and \(x\mapsto\mu(e,x)\) are homotopic to \(\operatorname{id}_X\) through basepoint-preserving maps. Composing these homotopies with \(i\) and \(r\) gives basepoint-preserving homotopies \[ a\longmapsto\mu_A(a,e)\simeq r i(a)=a, \] and similarly \[ a\longmapsto\mu_A(e,a)\simeq a. \] Thus \(A\) is an H-space with identity \(e\).