Mathematics – Algebraic Topology
Scientific paper
2006-11-06
Mathematics
Algebraic Topology
21 pages
Scientific paper
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial collapse of their associated simplicial complexes. Moreover, a simplicial collapse induces a collapse of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. Furthermore, this class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.
Barmak Jonathan Ariel
Minian Elias Gabriel
No associations
LandOfFree
Simple Homotopy Types and Finite Spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Simple Homotopy Types and Finite Spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Simple Homotopy Types and Finite Spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-617749