165 problems

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…

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 $C$ 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…

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

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

The minimality conjecture for A(n). If $n\geq 2$, then $\langle A(n)\rangle$ is a minimal nonzero Bousfield class. Moreover, every such $E$ is Bousfield equivalent to a finite wedg…

The quotient-isomorphism conjecture. The epimorphism $r'$ is an isomorphism.

Let $G$ be the compact Lie group and $EG$ the universal principal $G$-bundle, with $Ad(EG)$ its adjoint bundle and $T_v$ its vertical tangent bundle. Let $LBG^{-TBG}$ denote the as…