Structure of k-cycles on a k-dimensional Space

Details Structure of k-cycles on a k-dimensional Space Structure of k-cycles on a k-dimensional Space

2012-03-23

arxiv.org/abs/1203.5288v1

Mathematics

Algebraic Topology

6 pages

Scientific paper

Definition 1. A coordinate space is a finite dimensional real vector space V with a choice of dimV co-dimension one subspaces in general position. By general position we mean that the dimension of the intersection of any n of these hyperplanes is dimV - n. In this note we attach, in a topologically invariant manner, a finite length chain complex of finite dimensional coordinate spaces to any compact triangulable space. The top homology of the chain complex agrees with the top homology of the space and thus inherits an oriented matroid structure from the coordinate space structure on the top chain groups. In dimension two, this chain complex invariant can be refined to give a complete topological invariant for a class of two dimensional spaces called taut. Taut two complexes exist in every possible homotopy type of connected two complexes, and are characterized by being built out of surfaces glued to graphs by maximally efficient attaching maps. This paper was motivated by our attempt to understand the possible content of a lost manuscript by Ralph Reid at MIT circa 1970.

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