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ASYMPTOTIC DIMENSION AND GEOMETRIC DECOMPOSITIONS IN DIMENSIONS 3 AND 4

Part of: Topological manifolds Higher algebraic $K$-theory Low-dimensional topology Differential topology Global differential geometry

Published online by Cambridge University Press:  21 April 2025

H. CONTRERAS PERUYERO Open the ORCID record for H. CONTRERAS PERUYERO [Opens in a new window]  and
P. SUÁREZ-SERRATO Open the ORCID record for P. SUÁREZ-SERRATO [Opens in a new window]
H. CONTRERAS PERUYERO
Affiliation:
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México UNAM, Antigua Carretera a Pátzcuaro # 8701, Col. Ex Hacienda San José de la Huerta, 58089, Morelia, Michoacán, Mexico e-mail:[email protected]
P. SUÁREZ-SERRATO*
Affiliation:
Max-Planck-Institute for Mathematics, Bonn, Germany and Instituto de Matemáticas, Universidad Nacional Autónoma de México UNAM, Mexico Tenochtitlan
*
e-mail: [email protected]
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Abstract

We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to four when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most three. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most three. Thus, we obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and for closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition.

Keywords

asymptotic dimensionNovikov conjecturegeometric decompositionsAlexandrov spaces

MSC classification

Primary: 57R19: Algebraic topology on manifolds 19D50: Computations of higher $K$-theory of rings 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)
Secondary: 57M50: Geometric structures on low-dimensional manifolds 57N16: Geometric structures on manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 26
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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Footnotes

Communicated by Graeme Wilkin

HCP was supported by UNAM Posdoctoral Program (POSDOC). PSS thanks the Max-Planck-Institute for Mathematics in Bonn.

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