162 problems

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…

Rognes' localization conjecture. These maps should fit into a cofiber sequence in the stable category

Let $p$ be a prime, let $\operatorname{eo}_{p-1}$ be the proposed connective form of $EO_{p-1}$, and let $a_2,\dots,a_{p-1}$ be exterior generators with … Let…

Let $\mathit{tmf}$ denote the connective topological modular forms spectrum, let $\Sigma_3$ be the symmetric group on three letters, and let $\mathit{tmf}^{t\Sigma_3}$ denote its T…

Let $p$ be a prime and let $\ell$ be a generator of $\mathbb{Z}_p^\times$ when $p$ is odd, or of $\mathbb{Z}_2^\times/\{\pm 1\}$ when $p=2$. Write $Q(\ell)_{K(2)}$ for the $K(2)$-l…

Let $p$ be an odd prime, let $V(1)$ be the Smith–Toda complex, and let $X$ be a type-$2$ atomic spectrum. For a finite $p$-torsion atomic spectrum $X$, define $e(X)=k$ when…

Let $p$ be an odd prime, let $\mathcal C_2$ be the thick subcategory of finite $p$-local spectra of type at least $2$, and let $V(1)$ denote the Smith–Toda complex. Adams's conject…

Restriction equivalence conjecture. There is a notion of weak equivalence of semi-infinite spectra and a notion of weak equivalence of Hilbert spectra such that the restriction map…

For $p=2$, let $K(1)$ be the spectrum considered as an algebra over the completed connective complex $K$-theory spectrum $\widehat{KU}_2$, and let an opposite algebra structure mea…

Let $R$ be a ring spectrum and let $E$ be a commutative localized quotient of $R$. For a finite cell $R$-module $M$, let $\hat{\mathrm{L}}^R_E M$ denote the completed $E$-localizat…