The Seifert-van Kampen theorem and generalized free products of -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 | References | Similar Articles | Additional Information

Abstract: In a Seifert-van Kampen situation a path-connected space may be written as the union of two open path-connected subspaces and along a common path-connected 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.

**1.**A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May,*Rings, modules, and algebras in stable homotopy theory*, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR**1417719****2.**Paul G. Goerss and John F. Jardine,*Simplicial homotopy theory*, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. MR**1711612****3.**Daniel M. Kan,*A combinatorial definition of homotopy groups*, Ann. of Math. (2)**67**(1958), 282–312. MR**0111032****4.**Daniel M. Kan,*On homotopy theory and c.s.s. groups*, Ann. of Math. (2)**68**(1958), 38–53. MR**0111033****5.**Wilhelm Magnus, Abraham Karrass, and Donald Solitar,*Combinatorial group theory: Presentations of groups in terms of generators and relations*, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. MR**0207802****6.**William S. Massey,*Algebraic topology: an introduction*, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1967 edition; Graduate Texts in Mathematics, Vol. 56. MR**0448331****7.**Roland Schwänzl, Ross Staffeldt, and Friedhelm Waldhausen,*-theory and generalized free products of -algebras*, Geometry and Topology, to appear.**8.**Friedhelm Waldhausen,*Algebraic 𝐾-theory of generalized free products. I, II*, Ann. of Math. (2)**108**(1978), no. 1, 135–204. MR**0498807****9.**Friedhelm Waldhausen,*Algebraic 𝐾-theory of topological spaces. I*, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 35–60. MR**520492****10.**Friedhelm Waldhausen,*Algebraic 𝐾-theory of spaces*, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR**802796**, 10.1007/BFb0074449**11.**-,*On the construction of the Kan loop group*, Doc. Math.**1**(1996), No. 05, 121-126 (electronic).

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Additional Information

**Roland Schwänzl**

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

Email:
roland@mathematik.uni-osnabrueck.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/S0002-9939-02-06521-8

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