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A NOTE ON p-ADIC SIMPLICIAL VOLUMES

Part of: Other (nonclassical) types of functional analysis Homology and cohomology theories Topological fields Low-dimensional topology

Published online by Cambridge University Press:  13 August 2020

STEFFEN KIONKE  and
CLARA LÖH
STEFFEN KIONKE
Affiliation:
Fakultät für Mathematik und Informatik, Fernuniversität in Hagen, 58084Hagen, Germany, e-mail: steffen.kionke@fernuni-hagen.de
CLARA LÖH
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany, e-mail: clara.loeh@mathematik.uni-r.de
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Abstract

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We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of p-adic simplicial volumes. As the main examples, we compute the weightless and p-adic simplicial volumes of surfaces. This is based on an alternative way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod p and p-adic approximation of simplicial volume.

MSC classification

Primary: 55N10: Singular theory
Secondary: 12J25: Non-Archimedean valued fields 46S10: Functional analysis over fields other than ${bf R}$ or ${bf C}$ or the quaternions; non-Archimedean functional analysis 57M20: Two-dimensional complexes
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 3 , September 2021 , pp. 563 - 583
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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