10 problems

Let $M$ be an integral homology sphere, let $K(M)$ denote its Kauffman bracket skein module, and let $\overline{K(M)}$ be its completion. Homology-sphere $(1+t)$-torsion conjecture…

Fix a prime $p$, and for a group or space $X$ let $\beta_i^p(X)$ denote the dimension of $H_i(X,\mathbf{Z}_p)$. A homology 4-sphere is a 4-manifold with the homology of the 4-spher…

Anisotropy conjecture. For an arbitrary field $\mathbb{F}$, any $\mathbb{F}$-homology sphere is generically anisotropic over $\mathbb{F}$.

Let $Y_n=\#_n\Sigma(2,3,5)$, where $\Sigma(2,3,5)$ is the Poincaré homology sphere. A bounding four-manifold for $Y_n\#-Y_n$ is a four-manifold whose boundary is $Y_n\#-Y_n$. Unbou…

The algebraic $g$-conjecture. Every rational homology sphere has Lefschetz property.

Integral homology sphere conjecture. No $k$-Lê–Yomdin singularity with $k>1$ has an integral homology sphere link.

Let $\Delta$ be a simplicial sphere or a rational homology sphere. Its $h$-vector is subject to the $g$-theorem conditions: Dehn–Sommerville symmetry, unimodality up to the middle,…

Let $\Delta$ be a flag homology sphere. Write its $\gamma$-vector as $\gamma^\Delta=(1,\gamma_1,\gamma_2,\ldots)$. A simplicial complex is flag when its faces are exactly the cliqu…

For integers $d,i\geq 1$, let $\mathcal{HS}(i,d)$ be the family of $(d-1)$-dimensional homology spheres without missing faces of dimension greater than $i$. For…

Let $\Delta$ be a homology sphere. Let $k[\Delta]$ be its Stanley–Reisner ring, let $\Theta$ be a linear system of parameters, and define $L(\Delta)$ to be the set of pairs…