Universal coefficient theorem

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Problem 2.3.39 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 2.3 "Homology Theory"

(a)(b)

Let $R$ be a commutative ring with 1. Define an $R$ -homology sphere to be a Hausdorff topological space $X$ such that $H_{i}(X;R) \cong H_{i}(S^{n}; R)$ for some sphere $S^{n}$ .

(a)

Let $W$ be a compact connected $(n+1)$ -manifold and let $X = \partial W$ . Suppose that $H_{i}(W, \mathbb {Z}_{2}) = 0$ for all $i > 0$ . Prove that $X$ is a $\mathbb {Z}_{2}$ -homology sphere.

(b)

Give an example of $R$ and an $R$ -homology sphere which is a non-orientable connected compact manifold. Justify your answer by quoting appropriate theorems. You may quote standard computations without proof.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.