69 problems

Let $X$ be a smooth projective variety. Its fundamental group is big if the image of the fundamental group of every positive-dimensional subvariety in $\pi_1(X)$ is infinite. Kollá…

Singer–Hopf conjecture.

Let $d\geq 3$, and let $t_k^{\mathrm{perc}}$ be the critical threshold for homological percolation in degree $k$, while $t_k^{\mathrm{ec}}$ denotes the corresponding zero of the ex…

Let $G_A$ be a 2D or 3D grid domain with $A$ cells, and let $\mathtt{Cov}_{G_A}(A+1)$ denote the covering configuration space for $A+1$ agents. Write $\chi(G_A)$ for the Euler char…

Let $N^{2k}$ be a closed aspherical $2k$-manifold. Euler characteristic conjecture. One should have … This conjecture predicts the sign of the Euler characteristic of every closed…

Let $M$ be a $d$-dimensional complementary weak pseudomanifold, and let $\chi(M)$ denote its Euler characteristic. Euler characteristic conjecture. The Euler characteristic of $M$…

Let $M$ be a closed aspherical manifold of dimension $2m$, where $m$ is an integer, and let $\chi(M)$ denote its Euler characteristic. Hopf–Thurston conjecture. Every such manifold…

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…

Let $M$ be a closed, connected, oriented topological $n$-manifold, with simplicial volume $||M||$ and Euler characteristic $\chi(M)$. Say that $M$ is aspherical when its universal…

Hopf–Chern–Thurston conjecture. One has

Aluffi's conjecture. The Euler characteristic of $Y_G$ satisfies

Smillie–Vogtmann's conjecture. The rational Euler characteristic $\chi(\operatorname{Out}(F_n))$ is always negative, and its absolute value grows faster than exponentially as a fun…

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…

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…

Sparla's conjecture. The inequality

A Calabi–Yau threefold is a three-dimensional Calabi–Yau variety, and its Euler number is denoted by $e(Y)$. Finiteness conjecture. There are only finitely many possible numbers th…

Let $X$ be a Gorenstein orbifold admitting a crepant resolution $\pi:Y\rightarrow X$, meaning that $\pi^*K_X=K_Y$. The crepant resolution conjecture. The orbifold Euler characteris…

Flag complex conjecture.

Euler characteristic conjecture.

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…

Absolute and relative $L^2$-Euler characteristic conjecture. The uniform coarse index of $D^\chi_{\mathrm{abs/rel}}$ computes an appropriate absolute or relative $L^2$-Euler charac…

Let $X$ be a complex projective $n$-fold with large fundamental group, meaning that the image of the fundamental group of every closed irreducible positive-dimensional subvariety i…

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…

Gromov's conjecture. If