A notion of geometric complexity and its application to topological rigidity

Erik Guentner; Romain Tessera; Guoliang Yu

doi:10.1007/s00222-011-0366-z

A notion of geometric complexity and its application to topological rigidity

Erik Guentner
, Romain Tessera
, Guoliang Yu

University of Hawaii
Vanderbilt University
CNRS UMR 5669, 'Unité de Mathématiques Pures et Appliquées' and project-team Inria NUMED, Ecole Normale Supérieure de Lyon

Research output: Contribution to journal › Article › peer-review

68 Citations (Scopus)

Abstract

We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M×ℝ ⁿ is homeomorphic to N×ℝ ⁿ, for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n,K), for any field K.

Original language	English
Pages (from-to)	315-357
Number of pages	43
Journal	Inventiones Mathematicae
Volume	189
Issue number	2
DOIs	https://doi.org/10.1007/s00222-011-0366-z
Publication status	Published - 1 Aug 2012
Externally published	Yes

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AB - We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M×ℝ n is homeomorphic to N×ℝ n, for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n,K), for any field K.

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A notion of geometric complexity and its application to topological rigidity

Abstract

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