16 problems

Let $\tau$ be a collection of singularity types, let $X_\tau$ be the classifying space for $\tau$-cobordisms, let $\mathcal K_\tau$ be the Kazarian space, let $\mathcal K^+_\tau$ b…

Let $M^n$ be a closed $n$-manifold. A marked cubical decomposition of $M^n$ is a cubical decomposition together with the marking data used to distinguish its cubical cells, and two…

Universality conjecture. The formal group ${\frak F}(x\otimes 1,1\otimes x)$ is the universal object in the category of formal groups over a (topological) Hopf algebra.

Let $W_0$ and $W_1$ be closed, connected, oriented $3$-manifolds. A rational homology ribbon cobordism from $W_0$ to $W_1$ is a ribbon cobordism $W$ from $W_0$ to $W_1$ such that t…

Let $l$ be an integer, and let $\overline{A_2}(l+1)$ and $\overline{\Sigma}^2(l)$ denote the corresponding singularity loci. Their Segre–Stiefel–Whitney classes are defined in the…

Glazyrin–Pak cobordism conjecture. For every integral curve $\gamma$, there is a unit rhombus $\rho$ and a dome over $\gamma\cup\rho$.

Let $\mathcal{C}_0$ be the decoration data for an $E_1$-monoidal $(\infty,n-1)$-category $\mathscr{F}_{n-1}(\mathcal{C}_0)$ of singular manifolds in $\mathbb{R}^n$, framed in codim…

Cobordism conjecture. This cobordism is oriented if $M$ and $N$ are oriented, the codimension $k$ is odd, and $r$ is even.

Ribbon rational homology cobordism conjecture. If there exists a ribbon $\mathbb{Q}$-homology cobordism from $Y_0$ to $Y_1$ and one from $Y_1$ to $Y_0$, then $Y_0$ and $Y_1$ are di…

Oriented-boundary conjecture. The disjoint union of two copies of $M$ is an oriented boundary.

Let $M$ and $M'$ be closed manifolds of the same dimension. They are cobordant if there exists a manifold with boundary $W$ such that $\partial W=M\sqcup M'$, and $M$ is null-cobor…

Let $S$ be a smooth projective surface with $b_1(S)=0$, $p_g(S)>0$, $K_S\ne0$, and only Seiberg–Witten basic classes $0$ and $K_S$. Let $H,c_1,c_2$ be chosen so that there are no r…

A compact Nash set is a compact semialgebraic set endowed with its Nash structure. Two compact Nash sets are strongly asym-Nash cobordant if they are related by the strong form of…

Let $(\mathscr{C}_1,\iota_1,\varphi_1)$ and $(\mathscr{C}_2,\iota_2,\varphi_2)$ be framed flow categories. Their associated stable homotopy types are the stable homotopy types dete…

Cobordism rationality and functional equation conjecture. The function $Z_{\Omega_\ast,\beta}^{X}(q)$ is a rational function in $q$; more precisely, every $f_I(q)$ is the Laurent e…

Fold-map cobordism conjecture. There exists a $1$-dimensional linear subspace $\mathfrak G^1\leq\mathfrak N_n$ such that, if $M$ admits a fold map into a stably parallelizable…