Alegbraic-Topology

Homology sphere A homology sphere is a (closed connected oriented) $n$-mainfolds with the same homology as the $n$-sphere. Poincaré first conjectured that any $n$-manifolds homologous (i.e. having the same homology) to the $n$-sphere should be homeomorphic to the $n$-sphere, then later he found a counterexample, which led him to a modified conjecture (which is the Poincaré’s conjecture). The counterexample he found is very interesting, one of the way to construct it is the folowing:...

Using fields to detect integral homology? So in the universal coefficient theorem, by taking $R = \mathbb{Z}$, we know that there is a short exact sequence (which splits): $0 \rightarrow H_n(C_{*}) \otimes_R M \rightarrow H_n(C_{*} \otimes_R M) \rightarrow Tor^R_1(H_{n-1}(C_*), M) \rightarrow 0$ Some interesting cases are when $M = \mathbb{Z}_{p} \text{ or } \mathbb{Q}$ , do the homology over $\mathbb{Q}$ and $\mathbb{Z}_p$ tells us something about the integral homology? In particular, if $\tilde{H}(X; \mathbb{Q}) = 0$ and $\tilde{H}(X; \mathbb{Z}_p) = 0$ for all $p$, does it imply that $\tilde{H}(X) = 0$ ?...