Hatcher-Algebraic-Topology · Problem 24

Cohomology

Mathematics — Graph Theory · homology-closed-3-manifolds

Problem

Let MM be a closed connected 33-manifold, and write H1(M;Z)H_1(M;\mathbb{Z}) as Zr⊕F\mathbb{Z}^r\oplus F, the direct sum of a free abelian group of rank rr and a finite group FF. Show that H2(M;Z)H_2(M;\mathbb{Z}) is Zr\mathbb{Z}^r if MM is orientable and Zr−1⊕Z2\mathbb{Z}^{r-1}\oplus\mathbb{Z}_2 if MM is nonorientable. In particular, r≥1r\geq 1 when MM is nonorientable. Using Exercise 6, construct examples showing there are no other restrictions on the homology groups of closed 33-manifolds. [In the nonorientable case consider the manifold NN obtained from S2×IS^2\times I by identifying S2×{0}S^2\times\{0\} with S2×{1}S^2\times\{1\} via a reflection of S2S^2.]

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