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

Authors: Roland Schwänzl and Ross Staffeldt
Journal: Proc. Amer. Math. Soc. 130 (2002), 3193-3208
MSC (2000): Primary 19D10, 55P43
Published electronically: May 8, 2002
MathSciNet review: 1912997
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Abstract: 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 $W$. The 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 $\operatorname{Nil}$-term.

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Additional Information

Roland Schwänzl
Affiliation: Fachbereich Mathematik/Informatik, Universität Osnabrück, 46069 Osnabrück, Federal Republic of Germany

Ross Staffeldt
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003

Received by editor(s): June 20, 1999
Received by editor(s) in revised form: June 15, 2001
Published electronically: May 8, 2002
Additional Notes: The second author was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 343, Bielefeld, Germany
Communicated by: Ralph Cohen
Article copyright: © Copyright 2002 American Mathematical Society