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Abstract:
A cubical complex is a geometric object that can be decomposed into
hypercubes of various dimensions. The simplest example of a
d-dimensional cubical complex is the boundary of a
(d+1)-dimensional cube, which gives a cubical decomposition of a
d-dimensional sphere. In particular, the boundary of a
(d+1)-dimensional cube has 2d+1 vertices. In this
talk, we will give a precise combinatorial definition of cubical complexes
and prove that any cubical decomposition of a d-sphere requires at
least 2d+1 vertices.
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