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?
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Sign up to join this communityThese two very recent surveys state the conjecture as open:
Robert Kropholler, Zeeman’s collapsibility conjecture (2013)
A. P. M. Kupers, Zeeman's conjecture (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