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Topological nonrigidity of nonuniform lattices
Author(s) -
Chang Stanley S.,
Weinberger Shmuel
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20151
Subject(s) - citation , mathematics , combinatorics , library science , computer science , mathematical economics
We say that an arbitrary manifold (M, ∂M) is topologically rigid relative to its ends if it satisfies the following condition: If (N , ∂N ) is any other manifold with a compact subset C ⊂ N for which a proper homotopy equivalence h : (N , ∂N ) → (M, ∂M) is a homeomorphism on ∂N∪(N\C), then there is a compact subset K ⊂ N and a proper homotopy ht : (N , ∂N ) → (M, ∂M) from h to a homeomorphism such that ht and h agree on ∂N∪(N\K ) for all t ∈ [0, 1]. We say that a manifold M without boundary is properly rigid or absolutely topologically rigid if we eliminate the requirements that h be a homeomorphism on ∂N ∪ (N\C) and agree with ht on ∂N ∪ (N\K ) for all t ∈ [0, 1]. Along the lines of the classical Borel conjecture that all closed aspherical manifolds are topologically rigid, Farrell and Jones [8] provide the following important theorem: