10
$\begingroup$

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?

$\endgroup$

1 Answer 1

15
$\begingroup$

These two very recent surveys state the conjecture as open:

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

$\endgroup$
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.