, Volume 8, Issue 1, pp 117-125
Date: 08 Aug 2012

Comparing minimal simplicial models

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We compare minimal combinatorial models of homotopy types: arbitrary simplicial complexes, flag complexes and order complexes. Flag complexes are the simplicial complexes which do not have the boundary of a simplex of dimension greater than one as an induced subcomplex. Order complexes are classifying spaces of posets and they correspond to models in the category of finite T 0-spaces. In particular, we prove that stably, that is after a suitably large suspension, the optimal flag complex representing a homotopy type is approximately twice as big as the optimal simplicial complex with that property (in terms of the number of vertices). We also investigate some related questions.

Communicated by Graham Ellis.
Research supported by the Centre for Discrete Mathematics and its Applications (DIMAP), EPSRC award EP/D063191/1.