318 problems

Ravenel's telescope conjecture. The functors

Curtis conjecture. Only the Hopf invariant one and Kervaire invariant one elements survive under $h$.

Freyd's Generating Hypothesis for ${\mathcal{F}}_p$. The functor $\pi_*$ is faithful. Therefore, ${\mathcal{M}}_p$ is a Freyd category, with ${\mathcal{F}}_p$ as its full subcatego…

Let $M$ denote the classical Mahowald invariant, and let $v_{n-1}$-periodic and $v_n$-periodic families be families in the classical stable homotopy groups organized by the corresp…

Let $p$) be a prime, and let $L_h$ denote chromatic localization at height $h$. The $K(n)$-local sphere is expected to exhibit a precise transchromatic decomposition under the loca…

Margolis Uniqueness Conjecture. If there is an exact equivalence

Greenlees's conjecture. This map is an isomorphism for all abelian compact Lie groups $A$.

Let $G$ be a finite group. The Segal conjecture. After completion at the augmentation ideal, the Burnside ring of $G$ and the cohomotopy of $BG$ should agree. This conjecture relat…

Let $p$ be an odd prime, let $C/\overline{\mathbb{F}}_p$ be a supersingular elliptic curve, and let $Q(\ell)$ be the totalization of the semi-cosimplicial spectrum … Write…

Over $\mathbb{C}$, let $\eta^{-1}\mathbb{S}$ be the η-periodic motivic sphere, let $P$ and $v_n$ denote the classes on the $E_2$-page of its motivic Adams spectral sequence, and le…

Morel's stable $\mathbb{A}^1$-connectivity conjecture. Under these hypotheses, $L_{\operatorname{mot}}E$ is $n$-connective.

Let $M(\alpha)$ denote the Mahowald invariant of a 2-primary stable homotopy class $\alpha$, and let $\nu_n$-periodic and $\nu_{n+1}$-periodic families be periodic families in the…

Let $C$ be the category appearing above, let $L{\mathsf{Sp}}$ be the corresponding localized category of spectra, and write $\partial^{BB}_\ast$ for the Goodwillie derivative funct…

Mahowald's conjecture. There is a $2$-adic equivalence

For a non-negative integer $k$, let $e_0\mathbf{g}^k$ denote the corresponding subset of Ext. Non-emptiness conjecture. The set $e_0\mathbf{g}^k$ is non-empty if and only if … for…

Let $n\geq 3$ be odd, and let $\Sigma_Q$ be the homotopy sphere obtained as the boundary of the manifold $Q$ in the relevant bordism group. Its class $[\Sigma_Q]$ lies in…

Let $k$ be a field of characteristic zero. Write $K^M_2(k)$ for the second Milnor $K$-theory group and $\pi_{1,0}^k$ for the motivic stable homotopy group in weight zero. Morel's…

Eccles conjecture. Under these hypotheses, the stable adjoint of $f$ either is detected by homology or is detected by a primary operation in its mapping cone.

Let $\mathcal{S}$ be the stable homotopy category, and let $\mathsf{BL}(\mathcal{S})$ denote its Bousfield lattice. A localizing subcategory is a full subcategory closed under the…

Let $L_1$ and $L_2$ be links, let $L_1\amalg L_2$ be their disjoint union, and let $\mathcal{X}_\mathit{Kh}(L)$ denote the Khovanov homotopy type of a link $L$. Disjoint-union conj…

Let $L$ be a link, let $m(L)$ denote its mirror, and let $\mathcal{X}_\mathit{Kh}^j(L)$ denote its Khovanov homotopy type in quantum grading $j$. Mirror duality conjecture. The spa…

Let $p$ be a prime, and let $e$ and $k$ be positive integers. Consider groups of the form $\mathbb Z_{p^e}\times\mathbb Z_{p^k}$, together with their $ku$-toral and complex bordism…

Dwyer–Mahowald–Kuhn conjecture. The displayed isomorphism on homotopy groups is induced by an equivalence of spectra

Let $X$ be a finite spectrum of type $n-1$, let $L_n$ denote localization at the $n$th monochromatic level, let $X_p^{\wedge}$ denote the $p$-completion of $X$, and let $t$ denote…

Localize all spaces at $3$. Let $B_2(27,35)$ denote the space occurring in the proposed spherical resolution, and let $\alpha_2$ be the element generating…