28 problems

Let $R_p$ be the cofiber of the transfer map $B\Sigma_p\to S^0$, and let $\operatorname{eo}_{p-1}$ be the proposed connective spectrum associated to $EO_{p-1}$. The higher-prime tr…

Let $R_p$ be the cofiber of the transfer map $B\Sigma_p\to S^0$, and let $\operatorname{eo}_{p-1}$ be the proposed connective spectrum associated to $EO_{p-1}$. The indecomposabili…

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 $k$ be the base field and let ${\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ be the Morel–Voevodsky homotopy category of $S^1$-spectra. For an integer $d$, let…

Let $f:X\to\mathbf A^1$ be a function on a smooth complex algebraic variety, and let $x$ be a closed point of $f^{-1}(0)$. Suppose the singular locus of $f$ is a curve with local c…

Let ${\bf X}$ and ${\bf Y}$ be finite spectra, and let $\mathcal Z^\infty({\bf X},{\bf Y})$ denote the set of maps from ${\bf X}$ to ${\bf Y}$ that induce the trivial homomorphism…

Let $p$ be an odd prime. Write $ku$ for connective complex K-theory, and let $BP\langle 1\rangle$ denote the corresponding truncated Brown–Peterson spectrum. A core of a spectrum i…

Let $\d$ be the spatial dimension, let $H$ be an internal symmetry group, and let $\mathcal{S}^{d}(H)$ denote the set of $(d+1)$-dimensional gapped short-range entangled Hamiltonia…

Equality conjecture. Some authors conjectured that

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…

RZ-X conjecture. The partially ordered sets are isomorphic:

Let $k$ be a positive integer, and consider the cardinalities of $k$-spectra $\operatorname{ScatFact}_k(w)$ of strictly balanced binary words of length $2k$. The first gap is the i…

Let $L(n)$ denote the $n$-th layer, defined by … Let $S^0$ be the sphere spectrum and $H\mathbb{Z}$ the Eilenberg–MacLane spectrum of the integers. The spectra $L(n)$ are spacelike…

Let $j=2^{s_1}+2^{s_2}+\cdots+2^{s_k}$ with $0\leq s_1<s_2<\cdots<s_k$ and $k>1$. Let $x_{2j}$ be the indicated exterior homology generator in $H_*(\underline{bo}_6)$, and let…

BoP truncation conjecture. Such spectra exist and satisfy all of the displayed irreducible splittings, the stable cofibration, and the stated cohomological and spectral decompositi…

Vasil'ev–Grechkoseeva conjecture. Suppose that $L$ is one of the following nonabelian simple groups:

Meta-Isomorphism Conjecture. The induced map

Let $D_2$ be the little $2$-disks operad equipped with its $S^1$-action, let $\Sigma^\infty_+$ denote suspension spectrum after adjoining a disjoint basepoint, and let $K$ be the d…

Let $\tau$ be a nonzero element of $\pi_{k-1}(\mathbb{S}^0)$, and let $C\tau$ denote its cofiber. Mahowald's conjecture. The spectrum $C\tau$ does not admit an $A_{\infty}$-structu…

Let $\mathbb{S}=\Sigma^\infty S^0$ be the sphere spectrum, and let $\tau_{[0,n]}\mathbb{S}$ denote its truncation to degrees from $0$ through $n$. Let $\mathcal{F}_L^{(n)}$ be the…

Let $BP\langle 2\rangle$ be the height-two truncated Brown–Peterson spectrum at $p=2$. Its space of units $GL_1(BP\langle 2\rangle)$ is an infinite loop space, with classifying spa…

Let $k$ be a commutative ring. Consider commutative $Hk$-algebra spectra, commutative monoids in spectra of simplicial $k$-modules, and commutative monoids in spectra of differenti…

Let $\mathtt{SW}^f$ be the Spanier–Whitehead category of finite spectra. The pointed $0$-sphere is $(S^0,\star)$, where $\star$ is the basepoint, and $((S^0,\star),0)$ denotes it i…

A cohomology theory is a cohomology theory defined on spectra, and a homology theory is a homology theory defined on spectra; two such theories have the same acyclics when they van…

The preceding constructions involve spectra $mathbf L(i)\wedge \mathbf T(2^{n-i}-1)$, together with the spectra appearing in the complex of Theorem; the final cofibration has sourc…