Non-zero degree maps
Here's one of my favorite topics in low-dimensional topology. Let $M, N$ be closed, oriented, connected $n$-manifolds. There's a lot of ways that $M$ and $N$ could be related: homeomorphic, cobordant, one could cover the other, etc. The relation I'm interested in today is called ``domination''. We say that $M$ dominates $N$ if there exists a non-zero degree map from $M$ to $N$. (Recall that a map $f$ is degree $d$ if $f_*([M]) = d[N]$.) For example, a covering map dominates with degree the number of sheets of the covering (up to sign). Here's a few cool facts about domination. If $f: M \to N$ is degree $d$, then: 1) The image of $\pi_1(M)$ is finite index in $\pi_1(N)$ (and is surjective if the degree is 1). 2) $rank(H_i(M)) \geq rank(H_i(N))$ for all $i$. 3) If $M$, $N$ are hyperbolic 3-manifolds, then $vol(M) \geq d \cdot vol(N)$ with equality if and only if $f$ is homotopic to a cover. (If you're familiar with the Gromov norm, the same ineq