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A notion of geometric complexity and its application to topological rigidity

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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×ℝn is homeomorphic to N×ℝn, 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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Author notes

  1. Romain Tessera

    Present address: UMPA, ENS de Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France

Authors and Affiliations

  1. Department of Mathematics, University of Hawai‘i at Mānoa, 2565 McCarthy Mall, Honolulu, HI, 96822, USA

    Erik Guentner

  2. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Romain Tessera & Guoliang Yu

Authors
  1. Erik Guentner
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  2. Romain Tessera
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  3. Guoliang Yu
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Correspondence to Romain Tessera.

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Guentner, E., Tessera, R. & Yu, G. A notion of geometric complexity and its application to topological rigidity. Invent. math. 189, 315–357 (2012). https://doi.org/10.1007/s00222-011-0366-z

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