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I was wondering if anyone knew of a reference for explicitly computing triangulations for small examples of 3-dimensional lens spaces (let's say excluding projective space and spheres). It needn't be minimal or anything like that: I'm looking for something to help my geometric intuition. In particular, I'd really love a picture of a triangulation using the bipyramid <--> lens space identification. Say, for example, a triangulation of L(3;1). I'm familiar with the usual CW-decomposition with a cell in every dimension, but I'd like a structure with finer granularity. If anyone has an intuitive way of constructing a triangulation in general, that would be even better!

Thanks so much!

squiggles
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    I don't know any reference, but for a triangulation you just need to check that your tetrahedra are "real" tetrahedra, i.e no vertex are glued together, etc. So you should take a big pyramid and subdivise enough such that all tetrahedra have at most one face/one edge/one vertex in commun with the boundary of the bipyramid, you should get what you want. –  Jul 19 '17 at 16:28

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This answer of mine may be helpful to you. A reference I found helpful with a pretty explicit decomposition is Turaev's book on torsions.

Neal
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  • Your pictures are excellent. Thank you so much. – squiggles Jul 20 '17 at 16:02
  • A follow up question: looking at your Heegard splittings, it's difficult for me to tell where the homologically nontrivial generators of the first homology are. Can you describe where these are? Intuitively it seems that moving along the equator the bipyramid should result in nontrivial loops, but it's difficult to visualize the obstruction to contracting them. – squiggles Jul 23 '17 at 00:29