Simple homotopy types and finite spaces

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• 作者：Jonathan Ariel Barmak ; Elias Gabriel Minian
• 关键词：Finite spaces ; Simplicial complexes ; Simple homotopy types ; Posets ; Weak homotopy equivalences ; Simple homotopy equivalences
• 刊名：Advances in Mathematics
• 出版年：2008
• 出版时间：1 May 2008
• 年：2008
• 卷：218
• 期：1
• 页码：87-104
• 全文大小：200 K

文摘

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 l1"">l1&_user=1067359&_cdi=6681&_rdoc=5&_acct=C000050221&_version=1&_userid=10&md5=ce1a024469daa3bf5c5d12ab627c2b47"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">XY of finite spaces induces a simplicial collapse of their associated simplicial complexes. Moreover, a simplicial collapse KL 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. 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.