18 problems

Let $P$ be a finite $p$-group, let $A(P,P)$ be its double Burnside ring, and let $H^*(P)$ be its mod $p$ cohomology ring. The algebra $A(P,P)\otimes{\mathbb F}_p$ acts on $H^*(P)$…

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…

Segal's conjecture. The completion homomorphism

Multiplicative Burnside conjecture.

Let $\mathcal{M}$ be a matroid of rank $r+1$ with Chow ring $A(\mathcal{M})=\bigoplus_{k=0}^r A_k$, let $G=\operatorname{Aut}(\mathcal{M})$, and let…

A variant of the Segal conjecture. On morphism sets, the functor $\mathbb{AG}\to\operatorname{Ho}(\mathord{\mathrm{Sp}})$ completes $\mathbb{AG}(G,H)$ at the augmentation ideal $I$…

Let $G$ and $H$ be finite groups. Let $A(G)$ be the Burnside ring of finite $G$-sets, let $I\subset A(G)$ be its augmentation ideal, and let $\mathbb{AG}(G,H)$ be the free abelian…

Let $G$ be a finite group and let $n\geqslant0$. The Ravenel ideal $\mathrm{J}_n(G)$ consists of virtual finite $G$-sets whose fixed-point sets have virtual cardinality zero for ev…

Let $G$ be a finite group, and let $\beta(G)$ denote its largest quotient that is a $B$-group. Thévenaz's conjecture. The group $\beta(G)$ is solvable if and only if $G$ is solvabl…

Let $G$ be a finite group and let $S^0_G$ denote the equivariant sphere spectrum in $\mathrm{Sp}_G$. Its Borel-completion is obtained by completing at the augmentation ideal in … w…

Deviation conjectures. With $\Delta=\Delta(G,H)$, one has $\Delta=1$ if and only if $H=G$, the index $[G:H]$ divides $\Delta$, and $\Delta$ divides $|G|$. These conjectures propose…

Conjecture B. The map $\nu_G$ is injective for every finite group $G$.

Let $p$ be a prime, let $G$ be a finite $p$-group, and let $H\cong \mathbb Z_p$. Write $\tilde G=G\times H$, let $N(G,H)$ denote the relative Burnside kernel, and for a subquotient…

Let $G$ be a finite group. Let $\Omega(G)$ be the Burnside ring, let $\Omega^*(G)$ denote its unit group, and let $\Omega_2(G)$ be the subring of monomial representations induced f…

Let $G$ be a compact Lie group and let $\mathcal F$ be a normal family of representations, consisting of representations $\psi_\alpha^H:H\to GL(V_\alpha)$ with $V_\alpha^H=\{0\}$ a…

Let $G$ be a compact Lie group and $X$ a $G$-space. Let $\Pi_0(G,X)$ be the component category whose objects are $G$-maps $x:G/H\to X$, and let $Is\,\Pi_0(G,X)$ be the set of isomo…

Let $G$ be a compact Lie group, let $f:Z\to V$ be an equivariant immersion of a closed oriented $G$-manifold $Z$ of codimension one, and let $w\neq 0$ be an invariant field in $V$.…

Let $G$ be a finite $p$-group, let $S_G$ be the genuinely $G$-equivariant sphere spectrum, and write … for the $G$-homotopy fixed points of a $G$-spectrum $X$. Graeme Segal's Burns…