17 problems

Intrinsic-bracket conjecture. The homomorphism $I$ induces an isomorphism of operads

Let $\Gmr$ be a finite group and let $p$ be a prime. For each $i$, let $\Ecal^{(p^i)}_{\Gmr,p}$ be the collection of integral $\uprho_{\Gmr}$-stable $\Hmr\underline{\FF}_p$-Euleria…

Let $n$ be a nonnegative integer. For a spin$^c$ manifold $M$, write $v_{4n}(M)$ for its degree-$4n$ Wu class, and let $Sq^3$ and $\beta^{\mathbb{Z}/2}$ denote the third Steenrod s…

Let $X^n$ be a space and let $p$ be a prime. Say that $H^*(X;\mathbb{Z}_p)$ is $k$-periodic up to degree $c$ when multiplication by a periodicity element induces successive isomorp…

Let $X$ be a topological space and let $p$ be an odd prime. A nonzero element $x\in H^k(X;\mathbb{Z}_p)$ induces periodicity if multiplication by $x$ gives isomorphisms … for all…

Let $X$ be a connected topological space. A nonzero element $x\in H^k(X;\mathbb{Z}_2)$ induces periodicity if multiplication by $x$ gives isomorphisms … for all $i\geq 0$; in this…

Let ${\mathtt P}=[n]$, and let $\overline{p}$ and $\overline{q}$ be Goresky–MacPherson perversities. Let ${\mathtt K}(R,n,{\mathtt P},\overline{p})$ denote the perverse Eilenberg–M…

Let $X$ be a stratified space, let $R$ be a commutative ring, and let $\overline{p}$ and $\overline{q}$ be Goresky–MacPherson perversities. Intersection cohomology operations are n…

Let $X$, $Y$, and $Z$ be finite sets with maps … Form the pullback square … For a map $m\colon W^2\to\mathbb{Z}$, let $\mu_m$ denote multiplication by the associated class $h_m$ on…

Primary, secondary, and tertiary cohomology operations are encoded by increasingly higher algebraic structures; in particular, the case $n=2$ concerns tertiary operations and $2$-t…

Let $k_2o\langle r\rangle$ denote the indicated connective twisted $KO$-theory spectrum, let $\underline{\mathcal A(1)}$ be the twisted Steenrod subalgebra, let $\varphi$ be the em…

Assume that $k$ contains all primitive $\ell^{th}$ roots of unity, and let $n\ge 2i$. Write … Call $IJ$ admissible when its excess is $e(IJ)$, and let $P^I$ and $P_V^J$ denote the…

Let $X$ be a space and let $n>2$. Assume that $LS_2(X)\leq n-2$, where $LS_2$ is the local LS-category invariant defined using an $n$-set cover whose pairwise unions map null homot…

Let $p$ be a prime, let $M$ be a topological space, and let $x\in H^k(M;\mathbb{Z}_p)$ be nonzero. Say that $x$ induces periodicity up to degree $r$ when multiplication by $x$ give…

Let $p$ be a prime, and let $\mathcal{B}$ be the bigraded differential algebra over $\mathbb{Z}/p^{2}$ that describes the algebra of cohomology operations of order $\leq 2$ up to e…

Let $\mathcal{B}_{(n)}$ denote the algebra of $(n+1)$-st order mod $p$ cohomology operations, with $n\geq 0$. For $n=0$, $\mathcal{B}_{(0)}=\mathcal{A}$ is the mod $p$ Steenrod alg…

Let $R=\mathbb{Z}/p^2$, and let $\mathcal{M}^N$ be the category of finitely generated free graded $R$-modules concentrated in degree less than $N$. For a bigraded differential alge…