The Seifert-van Kampen theorem and generalized free products of S-algebras

Autor(en): Schwanzl, R
Staffeldt, R
Stichwörter: Mathematics; Mathematics, Applied
Erscheinungsdatum: 2002
Herausgeber: AMER MATHEMATICAL SOC
Journal: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
Volumen: 130
Ausgabe: 11
Startseite: 3193
Seitenende: 3208
Zusammenfassung: 
In a Seifert-van Kampen situation a path-connected space Z may be written as the union of two open path-connected subspaces X and Y along a common path-connected intersection W. he fundamental group of Z is isomorphic to the colimit of the diagram of fundamental groups of the three subspaces. In case the maps of fundamental groups are all injective, the fundamental group of Z is a classical free product with amalgamation, and the integral group ring of the fundamental group of Z is also a free product with amalgamation in the category of rings. In this case relations among the K theories of the group rings have been studied. Here we describe a generalization and stablization of this algebraic fact, where there are no injectivity hypotheses on the fundamental groups and where we work in the category of S-algebras. Some of the methods we use are classical and familiar, but the passage to S-algebras blends classical and new techniques. Our most important application is a description of the algebraic K-theory of the space Z in terms of the algebraic K-theories of the other three spaces and the algebraic K-theory of spaces Nil-term.
ISSN: 00029939
DOI: 10.1090/S0002-9939-02-06521-8

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