6 problems

Stiefel's conjecture. The class $w_{n-i}(M)$ can be defined simply as the sum of all $i$-simplices in the barycentric subdivision of a triangulation of $M$.

Conjecture that $(-1)^{\int w_2^4}$ is a nontrivial QCA. The response $Y=(-1)^{\int w_2^4}$ is a nontrivial quantum cellular automaton.

Vanishing Stiefel–Whitney classes conjecture. The first $2^{t+1}-1$ Stiefel–Whitney classes of $M$ vanish if and only if, for every $S\subseteq\{1,2,\ldots,k\}$ with $|S|\leq t+1$,…

Let $M^n$ and $N^n$ be open connected smooth manifolds, each homotopy equivalent to a finite polyhedron, and let $f\colon M^n\to N^n$ be a continuous homeomorphism. Topological inv…

Let $S$ be a normal divisorial scheme over $\mathbb Z[\frac1{2\ell}]$, either separated or noetherian, and let $f:X\to S$ be a proper smooth morphism of relative even dimension $n$…

Let $X$ be a proper and smooth scheme of even dimension $n$ over a field $K$ of characteristic different from $2$ and $\ell$. For each integer $q\geq 0$, let $e_q:G_K\to\{\pm1\}$ b…