Application
If \(X\) a simply connected, closed 3-manifold is a homology sphere, then it is a homotopy sphere. ^8a317f
- \(H_0 X = {\mathbf{Z}}\) since \(X\) is path-connected
- \(H_1 X = 0\) since \(X\) is simply-connected
- \(H_3 X = {\mathbf{Z}}\) since \(X\) is orientable
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\(H_2 X = H^1 X\) by Poincaré duality. What group is this?
- \(0 \rightarrow \operatorname{Ext}_{{\mathbf{Z}}}^{1}\left(H_{0}(X ; \mathbb{Z}), \mathbb{Z}\right) \rightarrow H^{1}(X ; \mathbb{Z}) \rightarrow \operatorname{Hom}_{\mathbb{Z}}\left(H_{1}(X ; \mathbb{Z}), \mathbb{Z}\right) \rightarrow 0\) yields
- \(0 \rightarrow \operatorname{Ext}_{{\mathbf{Z}}}^{1}\left({\mathbf{Z}}, \mathbb{Z}\right) \rightarrow H^{1}(X ; \mathbb{Z}) \rightarrow \operatorname{Hom}_{\mathbb{Z}}\left(0, \mathbb{Z}\right) \rightarrow 0\)
- Then \(\operatorname{Ext}_{{\mathbf{Z}}}^{1}\left({\mathbf{Z}}, \mathbb{Z}\right) = 0\) because \({\mathbf{Z}}\) is a projective \({\mathbf{Z}}{\hbox{-}}\)module, so \(H^1 X = 0\).
- So \(H_*(X) = [{\mathbf{Z}}, 0, 0, {\mathbf{Z}}, 0, \cdots ]\)
- So \(h_3: \pi_3 X \to H_3 X\) is an isomorphism by Hurewicz. Pick some \(f\in \pi_3 X \cong {\mathbf{Z}}\). By partial application, this induces an isomorphism \(H_* S^3 \to H_* X\).
- Taking CW approximation for \(S^3, X\), we find that \(f\) is a homotopy equivalence.