Intuition

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 $\tilde{H}(X) = 0$ ?...

Warning You need to know some basic group theory terminology to appreciate(I hope you do) the following content. Burnside’s lemma Here is the statement of the Burnside’s lemma: Let $G$ be a group that actions on a set $X$. Denote $X^{g}$ the set of fixed points of $g$ i.e. $\{x \in X | g \cdot x = x\}$, then the number of orbits of the action is equal to $\dfrac{1}{|G|}\sum\limits_{g \in G} |X^g|$....