27 problems

Alternation conjecture. The Euler characteristic is alternating:

Flag complex conjecture.

Euler characteristic conjecture.

Let $(X,g)$ be a closed $2n$-dimensional Riemannian manifold with sectional curvature $\mathrm{sec}_{g}$. Write $\chi(X)$ for its Euler characteristic. Hopf conjecture. … and … Thi…

Let $W$ and $W'$ be coefficient points for the varieties $Y_{W,n}$ and $Y_{W',n}$, respectively, and let $E_A$ denote the principal $A$-determinant, whose factors are the discrimin…

Let $M$ be a closed real $2n$-manifold. It is aspherical when its universal covering is contractible. Chern–Hopf–Thurston conjecture. If $M$ is aspherical, then it satisfies … This…

Hopf-Chern conjecture.

Let $X$ be a closed, oriented, aspherical $4$-manifold, and let $\tau(X)$ denote its signature. Gromov–Lück inequality. One has … This inequality is a refined form of the Hopf prob…

Singer–Hopf conjecture.

Let $X$ be a compact Kähler manifold of complex dimension $n$, and write … where $\Omega_X^p$ is the sheaf of holomorphic $p$-forms. Assume that $X$ is aspherical or has a nef cota…

Let $m\geq1$, let $s\in\{m+3,\ldots,2m+3\}$, and set $n=2s-2m-3$. Let $\mathsf{M}_{m,s}$ be the degeneracy locus considered in the paper and let $Z_{n,s,m}$ be the associated varie…

Let $X$ be an aspherical complex projective manifold and let $Z\subseteq X$ be a closed irreducible subvariety. Let $Eu_Z$ be MacPherson's local Euler obstruction function, let…

Aspherical 4-manifold Euler-characteristic problem. There exists an oriented closed aspherical $4$-manifold $M$ with

Arapura–Wang's conjecture.

Let $M$ be a compact $3$-manifold with prime decomposition … where the prime factors are as in (1.1) and categories (i)–(iv): the $M_i$ are closed prime $3$-manifolds with infinite…

Let $X$ be an aspherical projective manifold and let $\mathcal{P}$ be a perverse sheaf on $X$. Perverse-sheaf Singer–Hopf conjecture. The Euler characteristic of $\mathcal{P}$ is s…

Weil conjecture. The $\ell$-adic component of $\chi(V(K_1))$ is

Let $M^{3}$ be a closed, oriented Haken $3$-manifold. A compact, oriented Haken $4$-manifold $W^{4}$ is required to have boundary $\partial W^{4}=M^{3}$, with boundary understood t…

Hopf–Thurston sign conjecture. The Euler characteristic of $M$ satisfies

Let $I$ be a finite category with series Euler characteristic. Write $A_I$ for its adjacency matrix and $\chi_{\Sigma}(I)$ for its series Euler characteristic. The zeta function of…

Noguchi's conjecture. The zeta function is a finite product of the form

Let $M$ be a closed oriented aspherical manifold. Its tangent bundle is denoted by $TM$, and a flat structure on $TM$ means that it is induced by a representation of the fundamenta…

Let $w$ be a minuscule web with boundary $\vec{\lambda}$ and dual diskoid $D$. Let $F(\vec{\lambda})$ be the Satake fibre, let $Q(D)$ be the diskoid-configuration variety, and let…

Let $w$ be a closed web, meaning a web with no boundary, and let $D$ be its dual diskoid. Let $Q(D)$ be the variety of diskoid configurations associated with $D$, let $\Psi(w)$ den…

Sparla's conjecture. The inequality