89 problems

Let $X$ be a finite-dimensional topological space, and let $T^k=(S^1)^k$ be a $k$-dimensional torus acting almost freely on $X$. The total Betti number is…

Benson–Carlson conjecture. Any finite group of rank $r$ acts freely on a finite CW-complex homotopy equivalent to

Let $K$ be a convex body in $\mathbb R^d$, let $p$ be a prime, and let $k$ be a positive integer, and set $n=p^k$. Nandakumar–Ramana Rao theorem. It is possible to partition $K$ in…

Let $X$ be a compact Hausdorff space with a continuous free action of a nontrivial compact Hausdorff group $G$. Let $X*G$ denote the join equipped with the diagonal action of $G$.…

Let $C_5$ be the $5$-cycle and let $K_n$ be the complete graph on $n$ vertices. The 5-cycle complex $\operatorname{Hom}(C_5,K_n)$ is the Lovász complex of graph multimorphisms from…

Conner conjecture. The orbit space $X/G$ has trivial reduced Čech cohomology.

Makeev's conjecture. There exists an $A\in SO(n)$ such that all $n(n+1)/2$ of these $(n-1)$-planes are concurrent. The conjecture is a reformulation of Makeev's universal-cover con…

Smith conjecture. The isotropy representation of $G$ at $p$ is isomorphic, as a representation over $\mathbb{R}$, to the isotropy representation of $G$ at $q$.

Let $M$ be a closed oriented manifold with an effective $S^1$-action. For each connected component $F$ of $M^{S^1}$, assume the hypotheses on $\Gamma_F$, its smooth subalgebra, and…

Let $M$ be as above, let $M^\prime\to M$ be the induced connected normal $\Gamma^\prime$-covering, and let $G^\prime$ be the group of pairs $(\phi,g)$ satisfying…

Let $G$ be a finitely generated group of covering translations acting on a Whitehead manifold $W$, with quotient a 3-manifold $M$. A proper plane $\Pi$ in $W$ is equivariant if, fo…

Let $D^{p+q}$ be a disk in $S^p\times S^q$, let $\pi^{m-1}_{eq}$ denote the indicated equivariant homotopy group, and let $\Pi^{m-1}_{p-1,q}$ be the comparison group used in the pa…

Let $C_5$ be the cycle graph on five vertices, let $K_{n+2}$ be the complete graph on $n+2$ vertices, and let $\operatorname{Hom}(C_5,K_{n+2})$ be its graph homomorphism complex. L…

Simmons and Su's conjecture. There must exist $k$ adjacent vertices in the triangulation whose labels are $\{\omega^j m:1\leq j\leq k\}$ for some fixed $m$.

Let $G$ be a group, let $\underline{A}(G)$ be its Burnside group, and let … be the homomorphism induced by sending a finite proper $G$-set $S$ to the finitely generated projective…

Let $G$ be an algebraic group over $\mathbb R$, let … and let $p$ be a prime number. Regard $G(\mathbb C)$ as a $\Gamma$-space and let $G(\mathbb C)^{\operatorname{top}}$ denote it…

Let $G$ be a compact Lie group acting on Euclidean $n$-space or on the closed $n$-disk. Conner's conjecture. The orbit space is contractible. Conner's conjecture was proved by Oliv…

Dovermann–Suh subgroup conjecture. If

Laitinen's intersection conjecture. One may conjecture that

Let $T^k$ be the $k$-torus and let $\mathbb{Z}_2^k$ be its real subgroup. Write $\Omega^{U,T^k}_{2*}$ and $\Omega^{U,\mathbb{Z}_2^k}_{2*}$ for the unitary equivariant bordism group…

Let $K$) be a $d$-dimensional convex body in $\mathbb{R}^d$, let $\mu$ be an absolutely continuous measure on $\mathbb{R}^d$, let $n\geq 2$ be a natural number, and let…

Let $K\subset S^3$ be an L-space knot, meaning that some non-trivial positive Dehn surgery on $K$ is an L-space. A knot is strongly invertible if it admits an orientation-preservin…

Let $(Y,\tau)$ be a real rational homology sphere equipped with a compatible real $\mathrm{spin^c}$ structure. The real monopole Floer homology $\mathit{HMR}^{\circ}$ is defined by…

Let $\mathbb{S}_{\mathrm{E}}^m$ denote the $m$-dimensional Euclidean sphere equipped with its standard $\mathbb{Z}_2$-action, and let $d_{\mathrm{GH}}^{\mathbb{Z}_2}$ be the…

Let $K$ be a strongly invertible knot, represented by an intravergent diagram $D_i$ or a transvergent diagram $D_t$. For the transvergent diagram, consider the Borel equivariant Kh…