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Recall the Zeeman's conjecture: if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.

Zeeman showed that this implies the Poincaré conjecture in dimension 3 (which is equivalent to a restricted version of Zeeman's conjecture). It also implies the Andrews-Curtis conjecture (again, there is equivalence of a restricted version). In advertising Zeeman's conjecture these seem to be the two main points. (For an example of such an 'advertisement' see these notes.)

Question 1: What are some other interesting consequences of Zeeman's conjecture being (or not being) true?

I have seen an opinion that the conjecture is generally believed to be false.

Question 2: What are the reasons to believe it is false?

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    $\begingroup$ Regarding Question 2: there seem to be unpublished counterexamples to the Andrews-Curtis conjecture: people.maths.ox.ac.uk/kar/Past%20classes.html $\endgroup$
    – ThiKu
    Commented Jun 18, 2014 at 4:31
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    $\begingroup$ That doesn't seem to be an accurate summary of what you find on that web page. There is a non-specific reference to an unpublished result of Bridson, and a reference to a published paper of Burns-Macedonska giving potential counterexamples. $\endgroup$
    – Danny Ruberman
    Commented Jun 20, 2014 at 22:39

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