In this talk I will do an unusual thing in discussing a project that did not work, as it provides some insight into why the Poincaré Conjecture is difficult!

The talk is primarily concerned with studying simplicial maps with contractible point inverses and when they are homotopic to piecewise linear (p.l.) homeomorphisms. This will be related to the (strong) p.l. Poincaré Conjecture that asserts that every p.l. homotopy sphere is a p.l. sphere. I will also explain what can actually be proved with these techniques.

I am to make the talk very accessible and there will be lots of pictures!

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