Suppose that a CW complex \(X\) contains a subcomplex \(S^1\) such that the inclusion \(S^1 \hookrightarrow X\) induces an injection \(H_1(S^1; \mathbb{Z}) \to H_1(X; \mathbb{Z})\) with image a direct summand of \(H_1(X; \mathbb{Z})\). Show that \(S^1\) is a retract of \(X\).
Solution
Let \[ i:S^1\hookrightarrow X \] be the inclusion. By hypothesis its image in \(H_1(X;\mathbb Z)\) is a direct summand isomorphic to \(\mathbb Z\). Hence there is a homomorphism \[ \lambda:H_1(X;\mathbb Z)\to\mathbb Z \] whose composite with \[ i_*:H_1(S^1)\to H_1(X) \] is the identity.
The universal coefficient theorem gives \[ H^1(X;\mathbb Z)\cong\operatorname{Hom}(H_1(X),\mathbb Z). \] Let \(u\in H^1(X;\mathbb Z)\) correspond to \(\lambda\). By Theorem 4.57 choose \[ r:X\to S^1 \] with \(r^*(\iota)=u\), where \(\iota\) generates \(H^1(S^1)\). Then \[ (ri)^*\iota=i^*u=\iota, \] so \(ri:S^1\to S^1\) has degree \(1\) and is homotopic to the identity.
Since \(S^1\subset X\) is a subcomplex, the homotopy extension property extends a homotopy \[ ri\simeq\operatorname{id}_{S^1} \] to a homotopy of \(r\). At the endpoint we obtain \(r':X\to S^1\) satisfying \[ r'i=\operatorname{id}_{S^1}. \] Hence \[ \boxed{S^1\text{ is a retract of }X.} \]