44 problems

For every non-empty compact Hausdorff space $X$, every non-trivial compact group $\mathbb{G}$, and every free continuous action $X\times\mathbb{G}\to X$, there is no continuous…

For a finite group $G$ and finite free $G$-spaces (or finite free $G$-simplicial complexes) $X$ and $Y$, the generalized topological Hedetniemi conjecture asserts that…

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 $G$ be a finite group, and let $\operatorname{prim}(G)$ denote its primitivity invariant and $\operatorname{Sm}(G)$ its Smith invariant. Laitinen's conjecture. If … then … The…

Let $K$ be a planar convex body and let $n>1$ be a natural number. A partition of the plane into $n$ convex pieces is a collection of convex pieces $P_1,\ldots,P_n$ whose union is…

Let $G$ be a compact group. A metric $G$-ANE space is a metric $G$-space $\Bbb X\in G$-$\operatorname{ANE}$. An arbitrarily fine domination of $\Bbb X$ by $G$-$\operatorname{CW}$-c…

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…

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 $G$ 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…

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 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 $T^k$ 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 $\mathbb{S}_{\mathrm{E}}^m$ 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 $M$ be a closed smooth manifold, let $f:M\to M$ be a self-map, and let $\ell_n(f)$ lie in the direct sum $\bigoplus_{k\mid n}\mathbb{Z}[\pi_1(M)_{\rho,k}]$, where…

Let $G$ act on a closed smooth manifold $M$, let $f:M\to M$ be a $G$-equivariant map, and let $\ell_G(f)$ be the Klein–Williams invariant. For each conjugacy class of subgroups $H$…

Let $X$ and $Y$ be $\mathbb{Z}_2$-spaces, with product equipped with the diagonal $\mathbb{Z}_2$-action, and let $\operatorname{ind}(X)$ denote the index of a $\mathbb{Z}_2$-space.…

Let $\mathbb Z_n$ act smoothly on $S^4$, and suppose the action has exactly two fixed points. Point-fixing linearization conjecture. The action is smoothly equivalent to a linear a…

Let $(J,\tau)$ be the strongly invertible $2$-knot referred to in Gabbard's Theorem 5.8. Gabbard's equivariant sliceness conjecture. The knot $(J,\tau)$ does not bound an equivaria…

Let $\mathbb Z_n$ act smoothly on $S^4$, and suppose its fixed-point set is an unknotted two-sphere. Unknot linearization conjecture. The action is smoothly equivalent to a linear…

Rognes's equivariant equivalence conjecture. There should be a $GL_k(R)$-equivariant equivalence

Let $F(\mathbb{R}^2,n)$ denote the ordered configuration space of $n$ distinct points in the plane, and let $\mathrm{TC}^{\Sigma_s}[F(\mathbb{R}^2,s+1)\to F(\mathbb{R}^2,s)]$ denot…

Equivariant chain-level isomorphism conjecture. The chain complexes $(C^\epsilon,\partial^\epsilon)$ and $(\boldsymbol{C},\boldsymbol{\partial})$ are isomorphic as $\mathbb{Z}[G]$-…

Let $X$ range over real $2n$-dimensional torus manifolds corresponding to a multifan $\mathcal S_X$. Let $h_k(\mathcal S_X)=\sum_{i=k}^n(-1)^{i-k}\binom{i}{k}e_{n-i}(\mathcal S_X)$…

Let $M^n$ be a manifold with a “nice” group action or a space with a “nice” stratification, and let $N^n$ be another manifold or space that is equivariantly or stratified homotopy…

Let $G$ be a locally compact group, let $K$ be a compact large subgroup of $G$, and let $S$ be a $K$-space. The neighborhood equivariant retract conjecture. $S$ is a neighborhood…