3 problems

Let $M$ and $N$ be closed orientable manifolds, and let $f:M\to N$ be a map of degree one. The Lusternik–Schnirelmann category $\operatorname{cat}(X)$ is the least integer $n$ such…

Finiteness conjecture. Every closed orientable $4$-manifold $X$ 1-dominates at most finitely many closed orientable $4$-manifolds $Y$ with finite $\pi_1(Y)$, up to homeomorphism.

Let $M$ and $N$ be closed orientable 3-manifolds. A degree-one map is a map $f\colon M\to N$ of degree one, and $g(M)$ denotes the Heegaard genus of $M$. The degree-one map conject…