On the Proalgebraic Fundamental Group of Topological Spaces and Amalgamated Products of Affine Group Schemes

Deninger, C. and Wibmer, M. (2026) On the Proalgebraic Fundamental Group of Topological Spaces and Amalgamated Products of Affine Group Schemes. In: Longo, M., Seveso, M.A., Venerucci, R. and Vigni, S., (eds.) Elliptic Curves and Modular Forms in Arithmetic Geometry. Elliptic Curves and Modular Forms in Arithmetic Geometry, Celebrating Massimo Bertolini’s 60th birthday, 12-16 Sep 2022, Milan, Italy. Springer Proceedings in Mathematics & Statistics, vol. 527. Springer, pp. 101-152. ISBN: 978-3-032-13122-5. ISSN: 2194-1009. EISSN: 2194-1017.

Abstract

The proalgebraic fundamental group of a connected topological space X, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces on X. In this article, we further develop the theory of the proalgebraic fundamental group, in particular, we establish homotopy invariance and a Seifert–van Kampen theorem. To facilitate the latter, we study amalgamated free product of affine group schemes. We also compute the proalgebraic fundamental group of the arithmetically relevant Kucharcyzk–Scholze spaces and compare it to the motivic Galois group.

Metadata

Item Type:	Proceedings Paper
Authors/Creators:	Deninger, C. Wibmer, M.
Editors:	Longo, M. Seveso, M.A. Venerucci, R. Vigni, S.
Copyright, Publisher and Additional Information:	This is an author produced version of a conference paper published in Elliptic Curves and Modular Forms in Arithmetic Geometry,  made available via the University of Leeds Research Outputs Policy under the terms of the Creative Commons Attribution License (CC-BY), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited.
Keywords:	Proalgebraic fundamental group; Amalgamated product of group schemes; Tannakian criterion for reducedness; Kucharczyk-Scholze space
Dates:	Accepted: 29 January 2026 Published (online): 1 May 2026 Published: 12 May 2026
Institution:	The University of Leeds
Academic Units:	The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds)
Date Deposited:	03 Feb 2026 15:04
Last Modified:	13 Aug 2026 15:03
Status:	Published
Publisher:	Springer
Series Name:	Springer Proceedings in Mathematics & Statistics
Identification Number:	10.1007/978-3-032-13123-2_5
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:237398

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On the Proalgebraic Fundamental Group of Topological Spaces and Amalgamated Products of Affine Group Schemes

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