Regular CW complex

For example, all the cell spaces, except for the trisps that we have defined up to now are regular CW complexes. Regular trisps are also regular CW complexes. 

Let X be a regular CW complex and a Z2-space, and denote the fixed point free involution on X by 7. As mentioned above, one can choose a simplicial structure on X such that 7 is a simplicial map. We define a Z2-map p X ... 

Just like A, this is a functor from the category of regular CW complexes to the category of posets, since a cellular map between regular CW complexes will induce an order-preserving map between their face posets. 

Given a regular CW complex K, we recall that the simplices of Bd K are exactly those sets bp, , bFt that correspond to sequences of closures of cells including each other Fi C 2 C C Fj. So we have... 

In particular, as an immediate corollary of (10.5) we see that an arbitrary regular CW complex is homeomorphic to the order complex of some poset. We also obtain an alternative way to define the barycentric subdivision of... 

When the set of cells of a CW complex is given by means of a combinatorial enumeration, and the cell attachment maps are not too complicated, for instance if the CW complex in question is regular, it is natural to attempt to use the standard notion of cellular collapse to simplify the considered topological space, while preserving its homotopy t3rpe. 

If X is a finitely dominated CW complex and X is a (regular) covering of X with group of covering translations n then the disjoint union X = XU pt.) is a finitely dominated CWrr-complex. The symmetric construction on X is written as... 

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