25 problems

Let $G$ be a group such that the classifying space for proper $G$-actions, $\underline{E}G$, has a finite model. Let $A_{\operatorname{ho}}(G)=\pi_G^0(\underline{E}G)$, let…

Strict cyclotomic Deligne conjecture. For every natural number $n$, the following diagram of genuine $S^1$-spectra commutes:

Let $G$ be a finite group and let $\mathrm{AN}^{RO}_G(\mathbb S)$ be the $RO(G)$-filtered Adams--Novikov object. Applying the totalization functor gives the filtered $G$-spectrum…

Multiplicative tom Dieck splitting conjecture. There exists a multiplicative structure $\Xi$ such that $\Xi$-algebras are in particular $G$-$E_{\infty}$ algebras and, in the augmen…

Assume $p\geq 5$. Let $E_{p-1}$ be the height-$(p-1)$ Morava $E$-theory spectrum with its $\mathbb{Z}/p$-action induced by the corresponding subgroup of the Morava stabilizer group…

Let $BP\mathbb{R}^{HHR}$ be the Hill–Hopkins–Ravenel spectrum, and let $T$ be the $\mathbb{Z}/p$-equivariant Borel-complete spectrum defined by the theorem giving a strict…

Let $BP\mathbb{R}^{HHR}$ denote the non-equivariant spectrum conjectured by Hill, Hopkins, and Ravenel, and let the spectra under discussion be the completed Johnson–Wilson-type sp…

Let $p>2$ be a prime, and let $BP^{\wedge(p-1)}$ denote the $(p-1)$-fold smash product of the Brown–Peterson spectrum. A $\mathbb{Z}/p$-equivariant structure on this spectrum is an…

Let $\mathrm{THR}(\mathbb{Z})$ denote the real topological Hochschild homology of the integers, regarded as a $\mathbb{Z}/2$-spectrum. Write $\mathrm{H}\\,\underline{\mathbb{Z}}$ f…

Untwisted multiplication conjecture. The maps constructed from the monoidal structure make $\widetilde{\Theta^n}F$ into a nonunital $E_\infty$-ring spectrum.

Let $G$ be a finite $p$-group, let $E$ be a $p$-complete complex-oriented spectrum with associated formal group of height $n$, and let $E^2(f)$ denote the induced map in degree-two…

Let $G=\mathbb{Z}/2$, and consider the classical Tate construction $t_{\mathbb{Z}/2}({\rm inf}^{\mathbb{Z}/2}_{e}(-))^{\mathbb{Z}/2}$. Davis and Mahowald found that it sends the…

Let $G$ be a finite group and $N$ a normal subgroup of $G$. Let $\mathscr{T}_{G,N}(-)$ denote the generalized Tate construction obtained from the relative geometric $N$-fixed point…

Classical blue-shift phenomenon. The spectrum $t_{G}({\rm inf}^G_{e}(E))^G$ is $v_{n-s_{G;E}}$-periodic for some positive integer $s_{G;E}$. When $s_{G;E}>n$, the $v_{n-s_{G;E}}$-p…

Consider the $\operatorname{\mathop{\rm Ext}}$ groups in the $rho$-Bockstein spectral sequence … Here, the power of $\tau$ is the exponent of $\tau$ in the relevant…

Let $p$ be a prime, let $A$ be a finite abelian $p$-group, and let $\mathcal{F}$ be a proper family of subgroups of $A$. Let $R$ be a commutative algebra regarded as a Borel-equiva…

Let $C_2$ be the cyclic group of order two, and let $\mathrm{tmf}_{C_2}$ be the homotopy pullback spectrum defined by the Tate diagram described above. A $C_2$-ring spectrum means…

The $\log_p$-conjecture. For all $n\geq r$, one has

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…

Cyclicity conjecture. The group

Let $G$ be a compact Lie group, let $\mathscr{T}=\operatorname{Ho}G\mathscr{S}$ be the equivariant stable homotopy category, and let $G\mathscr{C}$ be the thick subcategory generat…

Equivariant generating hypothesis. The restriction of $\underline{\pi}_*^G(-)$ to the subcategory of finite $G$-spectra is faithful. Equivalently, if a map $f:X\to Y$ between finit…

Let $G$ be a finite group. Write $\operatorname{\sf StHom}^{naive}(G)$ and $\operatorname{\sf StHom}(G)$ for the categories of naive and genuine $G$-spectra, respectively, and let…

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…

Let $G$ be a finite $p$-group and let $X$ be a bounded below $G$-spectrum of finite type mod$p$. Write $X^G$ for its $G$-fixed points and $X^{hG}$ for its $G$-homotopy fixed points…