24 problems

For coprime integers $(p,k)$, let $K_{p,k}$ be the dual knot of a simple $(1,1)$-knot in the lens space with lens-surgery parameter $(p,k)$, and let $Y_{p,k}$ be the homology spher…

Björner–Lutz conjecture. The triangulation poincare is the smallest triangulation of $\Sigma^3$ and is the unique triangulation of $\Sigma^3$ with this $f$-vector.

Let $\Delta$ be a homology $(d-1)$-sphere over a field $\mathbf{k}$, with Stanley–Reisner ring $\mathbf{k}[\Delta]=S/I_\Delta$. An l.s.o.p. is a linear system of parameters…

A prime homology 3-sphere is a prime 3-manifold whose homology is that of the 3-sphere. Finite-type invariant conjecture. Any two different prime homology 3-spheres are distinguish…

The $g$-theorem characterizes the $f$-vectors of boundary complexes of simplicial polytopes. For a homology sphere, its $g$-vector is the sequence associated with its $f$-vector, a…

Let $n$ and $r$ be integers with $r<n-1$, and let a continuous action of $SL(n,\mathbb{Z})$ be given on an $r$-dimensional mod 2 homology sphere. Mod 2 homology-sphere action conje…

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…

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

Unbounded handle complexity conjecture. Suppose $\Gamma_Y(i)<i/2$. Then the minimum number of $1$-handles and $2$-handles among four-manifolds that fill $\#_nY\#_n-Y$ grows without…

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…

Let $\Delta$ be a homology sphere, and let $\mathbf{k}[\Delta]$ be its face ring over a field $\mathbf{k}$. The hard Lefschetz theorem concerns the existence of a suitable Artinian…

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

Let $\Delta$ be a rational homology sphere. It has the strong Lefschetz property if there exist a linear system of parameters $\Theta$ for $\mathbb{Q}[\Delta]$ and a linear form…

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 $\Sigma$ be a homology sphere. Over an infinite field $\mathbb{F}$, its Stanley–Reisner ring is $A=\mathbb{F}[x_v\mid v\in V(\Sigma)]/I_\Sigma$. The complex $\Sigma$ is a Lefsc…

Let $\Sigma$ be a $(d-1)$-dimensional homology sphere, with $g$-vector $g(\Sigma)=(g_0,g_1,\ldots,g_{\lfloor d/2 \rfloor})$. A vector…

Let $\mathcal{C}$ be the topological concordance group of knots in $S^3$, let $\widehat{\mathcal{C}}$ be the corresponding concordance group of knots in homology spheres, and let…

Let $\Delta$ be a simplicial $d$-sphere, or more generally a simplicial homology $d$-sphere. The $h$-vector of $\Delta$ is required to satisfy the Dehn--Sommerville equations, and…

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…

Let $S$ be a $k$-neighbourly $k$-stacked $\mathbb{F}$-homology sphere of dimension $2k$, and let $m=f_0(S)$ be its number of vertices. Sigma-vector conjecture. One has … This is id…

A triangulated sphere is a simplicial complex homeomorphic to a sphere, and its $h$-vector is the integer vector $(h_0,h_1,\ldots,h_d)$ determined by its face numbers. An $M$-vecto…

Let $\mathcal{HS}(i,d)$ be the family of $d$-dimensional homology spheres without missing faces of dimension greater than $i$. Write $d+1=qi+r$ with $q\geq 0$ and $1\leq r\leq i$,…

Let $\mathcal{HS}(i,d,n)$ be the family of $d$-dimensional homology spheres with $n$ vertices and without missing faces of dimension greater than $i$. For integers $d\geq 0$ and…

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…