Zeeman's conjecture in topological combinatorics states that if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.


What is the status of this conjecture as of 2015? 



Status of Zeeman's collapsability Conjecture
These two very recent surveys state the conjecture as open:


* Robert Kropholler, [Zeeman’s collapsibility conjecture](https://web.archive.org/web/20150918223637/http://people.maths.ox.ac.uk/kar/Zeeman.pdf) (2013)
* A. P. M. Kupers, [Zeeman's conjecture](http://math.harvard.edu/~kupers/notes/zeemansconjecture.pdf) (2014)


It doesn't seem like there's been any progress since. In particular, the most important results towards the conjecture are:


(**Cohen**, 1975) If $K$ is a contractible $2$-polyhedron, then $K\times I^6$ is collapsible. Similarly, if $K$ is a contractible $n$-polyhedron for $n \geq 3$, then $K \times I^{2n}$ is collapsible.


(**Perelman**, 2003, *corollary to the Poincaré conjecture*) Zeeman's conjecture is true for standard $2$-polyhedra that are spines of $3$-manifolds


