16 problems

Alexander's conjecture. If $K$ and $L$ have a common subdivision, then they have a common stellar subdivision.

PL sphere-or-ball conjecture. If $[u,v]$ is full, then $(u,v)$ is a PL sphere of dimension $\rho(v)-\rho(u)-2$. Otherwise, $(u,v)$ is a PL ball of the same dimension.

Conjectured failure of regular-neighborhood invariance. The property fails in each of the following cases: (e) $M=S^{2k}$ and $K=\Sigma(S^{k-1}\sqcup S^{k-1})$, where $f$ and $g$ c…

Let $X$ be the polyhedral object and let the Fibering Lemma 7.1 refer to the fibering assertion established there for PL embeddings. The topological fibering conjecture. The Fiberi…

For $0<p<l$, let $KT^{3l+p}_{p,2l-1,AD}$ and $KT^{3l+p}_{p,2l-1,PL}$ denote the almost smooth and PL knotted-torus classification groups. The PL forgetful-map conjecture. The forge…

Let $N$ be a connected PL $n$-manifold and let $\alpha$ denote the Haefliger–Wu invariant of a PL embedding $N\to\mathbb{R}^m$. The refined Haefliger–Wu conjecture. The invariant…

Finite-type conjecture. Each arc complex of a connected bordered surface determines an underlying space of finite type $n<\finite$.

Let $Q$ be a finite polyhedron that is a $k$-dimensional PL manifold. A PL sphere is a PL manifold piecewise-linearly equivalent to the standard sphere of the same dimension. The P…

Let $C$ be the specified 2-pentachoron cut-open Cappell--Shaneson complex, and let $B^4_C$ be the topological 4-ball containing $C$ described in the paper. Standardness conjecture…

Let $X$ be an Alexandrov space, meaning a finite-dimensional complete length space with curvature bounded below in the Alexandrov sense, and suppose that $X$ contains no proper ext…

Let $\overline{\mathrm{B}\Gamma}_n^{\mathrm{PL}}$ denote the classifying space of codimension $n$ PL Haefliger structures with trivial normal bundle. Greenberg–Gelfand–Fuks conject…

Tame tangle manifolds are PL manifolds. The canonical PL structures of open or compact tame $m$-tangles restrict to PL $m$-manifold structures on tame tangle manifolds.

A thickening is a regular neighborhood of a PL submanifold. In codimension $q=1$, the known thickenings are $I$-bundles. Schoenflies conjecture. There are no codimension-one thicke…

A PL manifold is a topological manifold equipped with an atlas whose coordinate transformations are piecewise linear homeomorphisms. Two PL manifolds are homeomorphic if they are h…

Let $s,d,r\ge2$ be integers satisfying … and let $K$ be a finite $s$-dimensional simplicial complex. An almost $r$-embedding is a map $f:K\to\mathbb R^d$ such that…

Let $D=D^{s_1}\sqcup\ldots\sqcup D^{s_r}$ be a disjoint union of $r$ disks, let $f:D\to B^d$ be a proper PL map such that…