7 problems

Smooth Poincaré conjecture in dimension 4. There are no exotic $S^4$'s; equivalently,

Generalized Poincaré conjecture. Any homotopy $n$-sphere is homeomorphic to $S^n$.

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…

González-Acuña's conjecture. If $A$ is an Artin $n$-presentation such that $|A|=1$, then $A\sim T$. This is one of the conjectures relating Artin presentations of the trivial group…

The piecewise linear Poincaré conjecture. Every piecewise linear homotopy $4$-sphere is piecewise linear homeomorphic to $S^4$.

A homotopy 4-sphere is a closed $4$-manifold homotopy equivalent to the $4$-sphere $S^4$. Four-dimensional topological Poincaré conjecture. Every homotopy $4$-sphere is homeomorphi…

Poincaré-conjecture consequence. The monoid ${\mathcal M^{\mathrm{Top}}_{n}}$ never has non-trivial units, and ${\mathcal M^{\mathrm{PL}}_{n}}$ has no non-trivial units except pote…