The Seifertvan Kampen theorem and generalized free products of algebras
Authors:
Roland Schwänzl and Ross Staffeldt
Journal:
Proc. Amer. Math. Soc. 130 (2002), 31933208
MSC (2000):
Primary 19D10, 55P43
Published electronically:
May 8, 2002
MathSciNet review:
1912997
Fulltext PDF Free Access
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Abstract: In a Seifertvan Kampen situation a pathconnected space may be written as the union of two open pathconnected subspaces and along a common pathconnected intersection . The fundamental group of 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 is a classical free product with amalgamation, and the integral group ring of the fundamental group of is also a free product with amalgamation in the category of rings. In this case relations among the 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 algebras. Some of the methods we use are classical and familiar, but the passage to algebras blends classical and new techniques. Our most important application is a description of the algebraic theory of the space in terms of the algebraic theories of the other three spaces and the algebraic theory of spaces term.
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Additional Information
Roland Schwänzl
Affiliation:
Fachbereich Mathematik/Informatik, Universität Osnabrück, 46069 Osnabrück, Federal Republic of Germany
Email:
roland@mathematik.uniosnabrueck.de
Ross Staffeldt
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email:
ross@nmsu.edu
DOI:
http://dx.doi.org/10.1090/S0002993902065218
PII:
S 00029939(02)065218
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
