12 problems

Let $X$ be a path connected $CW$-complex with finitely generated homology. For $n>0$, let $f\in\pi_n^sX\simeq\pi_nQX$, and let … be the unstable Hurewicz homomorphism. Eccles conje…

Let $R$ be a graded Noetherian ring, and let $\mathcal{D}(R)$ be the relevant derived or stable category with sphere object $S$. Write $\mathcal{GH}$ for the generating hypothesis,…

Let $E:\mathbf{BornCoarse}\to\mathbf{C}$ be a coarse homology theory, let $X$ be a uniform bornological coarse space, and let $i_\alpha:M_\alpha\to M$ be the canonical inclusions a…

The generalized Segal conjecture. For any bounded below spectrum $X$, the Tate diagonal map... This is presented in the source as part of a theorem attributed to Nikolaus and Schol…

Segal's conjecture. The composite map $\mS\to\mS^{t\rC _2}$ is 2-adic completion. The conjecture was first stated by Graeme Segal and was later proved by Gunnar Carlsson, so it is…

Detection conjecture. The nonzero element $t$ is detected by the sixth algebraic transfer $Tr_6(\mathbb F_2)$.

Let $W_0,W_1\subset S^3\times[0,1]$ be embedded cobordisms between links $L_0\subset S^3\times\{0\}$ and $L_1\subset S^3\times\{1\}$, and let…

Let $L$ be an annular link, let $Kh_{\mathbb{A}_\mathfrak{q}}(L)$ denote its quantum annular homology, and let ${\mathcal X}^r_{\mathbb{A}_\mathfrak{q}}(L)$ be the associated spect…

Assume $p$ is sufficiently large with respect to $n$. Let $E$ be the height-$n$ theory in the source, let $I_{n-1}$ be the corresponding invariant ideal, and let $L_{n-1}$ and…

Let $\operatorname{Stable}_{BP_*BP}$ be the stable category of $BP_*BP$-comodules. For each $n$, let $L_n^f$ denote finite localization with respect to a finite type $n$-spectrum,…

Let $\mathtt{NSH}$ be the triangulated noncommutative stable homotopy category, and let $A,B$ be objects of $\mathtt{NSH}$. Write $\pi_*(A)$ and $\pi_*(B)$ for their graded homotop…

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…