5 problems

Let $N$ be a $k$-connected closed $n$-manifold, with $n$ even, $m=(3n+1-k)/2\ge n+3$, and with $H_{k+1}(N)$ free. Write $\widetilde N$ for the universal cover and let…

The generalized Segal conjecture. For any bounded below spectrum $X$, the Tate diagonal map... This is presented in the source as part of a theorem attributed to Nikolaus and Schol…

Segal's conjecture. The composite map $\mS\to\mS^{t\rC _2}$ is 2-adic completion. The conjecture was first stated by Graeme Segal and was later proved by Gunnar Carlsson, so it is…

Let $Z$ be a Hausdorff space, let $\omega\colon M\to Z$ be continuous, and let $P=(P_1,P_2)$ range over regular index pairs for an isolated invariant set $S$. The parameterized Con…

Homotopy free rank conjecture. For groups whose rank is greater than one, $\operatorname{rk} G=\operatorname{hfrk} G$.