I got the same counter-example from Peter Johnstone and Peter Freyd. PJ called it the dunce cap and described it as a triangle with the three edges identified, one in the reverse direction. PF described it in a more complicated way that made it easier to see it actually contractible. And the latter was really what I wanted to refute. When I saw PJ's example, I mistakenly thought it still had a 1-cycle. The smallest triangulation I found had 8 vertices, which is still too large to compute by hand with. The dual complex still has over 200 simplexes and that is just too many. If anyone knows a smaller triangulation, I would like to know it. Thanks to all who responded. Michael