165 problems

Let $p$ be a prime, and let a spectrum of fp-type $n$ mean a bounded below, $p$-complete fp-spectrum whose associated thick subcategory of finite spectra has type at least $n+1$. L…

Let $0 \le i \le n+1$. For $i \le n$, let $\mathcal D_i$ be the full subcategory of dualizable objects $X$ in $\Sp_{k,n}^{\dual}$ that are retracts of $Y \wedge Z$ for some dualiza…

Chromatic splitting conjecture. If $X$ is the $p$-completion of a finite spectrum, then a splitting $\gamma_n$ exists for all $n$.

Let $E_{n-1}$-local spectra be localized with respect to Morava $E$-theory, let $S_p^0$ be the $p$-complete sphere spectrum, and let … denote the exterior-algebra-type wedge of $R$…

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

For every prime $p$, let $A_*$ denote the mod-$p$ dual Steenrod algebra and let $\operatorname{GrComod}_{A_*}$ denote the full category of graded $A_*$-comodules. The conjecture as…

For every abelian compact Lie group $A$ and every non-equivariantly Landweber exact ring spectrum $E$, the genuine $A$-equivariant ring spectrum $MU_A\wedge_{MU}E$ is equivariantly…

For each height $n$, the category of spectra of fp-type $n$ is the thick subcategory generated by the truncated Brown–Peterson spectrum $\mathrm{BP}\langle n\rangle$; equivalently,…

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…

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 $C$ 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…

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 a very exotic sphere mean a homotopy sphere that does not bound a parallelizable manifold. The preceding results show that very exotic spheres exist in at least $99$ congruence…

Parity conjecture. If $(1-\epsilon)\alpha$ is non-zero, then $w$ is even.

Hovey–Palmieri conjectures. The following assertions hold:

Stolz–Teichner conjecture. The degree-$n$ topological modular forms of $X$ are

For an integer $n>4$, let $S^n$ denote the $n$-dimensional sphere, and say that it has a unique smooth structure when every smooth manifold homeomorphic to $S^n$ is diffeomorphic t…

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…

Principal-localizing-subcategory conjecture. Every localizing subcategory is the collection of $E$-acyclics for some spectrum $E$, and is therefore principal.

Sphere–projective-space conjecture. Their Bousfield classes are equal:

X(n)-adjacency conjecture. The spectra $X(n)$ and $X(n+1)$ are adjacent in the Bousfield lattice: if

Bousfield-class dichotomy conjecture. Then

The Dichotomy Conjecture. Every spectrum has either a finite local or a finite acyclic.

Finite detection conjecture for I. If $E\wedge I\neq 0$, then

The decomposition conjecture for D. One has