10 problems

Higher-dimensional torus conjecture. There exists a $T^2$-invariant contact structure $\xi$ on $T^{2n+1}$ such that

Translated-subgroup support conjecture. The support of $\pi_n(\partial B_{\epsilon}-D)$ is a union of translated subgroups of $\operatorname{Char}\pi_1(\partial B_{\epsilon}-D)$ by…

The conjecture. Under these hypotheses, the map

Let $A$ be a pointed type, and let $n,m$ be natural numbers. The Whitehead product induces a map … Classically, this operation equips $\pi_{\bullet}(A)$ with a graded quasi-Lie alg…

Let $n$, $p$, and $m$ be integers with $m\geq 2$ and $1\leq p\leq n-3$. Let $\mathcal{L}_{m\cdot p}^{n}$ denote the space of long links with $m$ components of dimension $p$ in…

Let $X$ be a simply-connected space of finite type, and let $\operatorname{rank}_p(\pi_n(X))$ and $\operatorname{rank}_{\mathbb{F}_p}(\operatorname{H}_n(\Omega X;\mathbb{F}_p))$ de…

Let $M$ be a manifold with almost non-negative sectional curvature, abbreviated ANSC. The fundamental group $\pi_1(M)$ acts on the homotopy group $\pi_2(M)$ by deck transformations…

Let $\pi_n^3$ denote the $2$-component of the homotopy group $\pi_n(S^3)$. Ravenel's conjecture. The $2$-component of $\pi_n(S^3)$ is non-trivial for $n>10$. This conjecture concer…

Let $X$ be a pointed space, let $n\geq 1$, and let $\widetilde{\theta}_{\iota}$ be the operation on homotopy groups induced by the distinguished class $\iota$. For $i,j\geq 1$, ide…

Let $K$ be a connected $2$-dimensional CW-complex with subcomplexes $K_1,\dots,K_n\subset K$ such that $K_1\cup\dots\cup K_n=K$ and $K_1\cap\dots\cap K_n=K^1$. Set $F=\pi_1(K^1)$ a…