A notion of geometric complexity and its application to topological rigidity

By E. Guentner, Romain Tessera, Guoliang Yu. Inventiones Mathematicae, 2010

130 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f1ec7c255506089016f713dd352532af7863ddb0

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory (2011, methodology): “Recall that by Guentner–Tessera–Yu [13, Theorem 3.0.1] and [12, Theorems 5.2.2], Theorem 7.5 applies to every (geometrically finite) subgroup of GLn(A), where A is a commutative ring.”
Fibred coarse embeddings, a-T-menability and the coarse analogue of the Novikov conjecture (2013, background): “5], or the weaker properties of finite decomposition complexity [10] and property A [32].”
Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and $$C^*$$C∗-algebras (2015, methodology): “These implications come via the use of controlled cutting-and-pasting, or Mayer– Vietoris, techniques pioneered by the third author [44] in the setting of asymptotic dimension, and developed in a more general context by the first and third authors in collaboration with Tessera [15].”