3 problems

Let $X$ be a closed $n$-dimensional Riemannian manifold with first Betti number $b_1(X)=n$, and let $D$ be a positive constant. Write $\operatorname{Ric}(g)$ for the Ricci curvatur…

Let ${\mathbb{T}}^n$ be the $n$-torus, let $f:{\mathbb{T}}^n\to{\mathbb{T}}^n$ be a map, and let $\tau$ be a free involution of ${\mathbb{T}}^n$. Nielsen–Borsuk–Ulam dichotomy conj…

Let $T^6=(\mathbb{R}/4\mathbb{Z})^6=[0,4]^6/\sim$, and partition its $4^6$ subcubes into four classes. Nonexistence conjecture. No partition of the subcubes of $[0,4]^6/\sim$ gives…