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The Seifert-Van Kampen Theorem and Generalized Free Products of S-Algebras

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The Seifert-Van Kampen Theorem and Generalized Free Products of S-Algebras PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 11, Pages 3193{3208 S 0002-9939(02)06521-8 Article electronically published on May 8, 2002 THE SEIFERT-VAN KAMPEN THEOREM AND GENERALIZED FREE PRODUCTS OF S-ALGEBRAS ROLAND SCHWANZL¨ AND ROSS STAFFELDT (Communicated by Ralph Cohen) 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 fun- damental 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 applica- tion 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. 1. Introduction Consider a situation (1.1) W / /X Y / /Z where the space Z is given as the union of two open subsets X and Y with inter- section W . If all spaces are path-connected and are given the basepoint w W , then the Seifert-van Kampen theorem [6, page 114] states that the diagram2 of fundamental groups (1.2) π1(W; w) /π1(X; w) π1(Y;w) /π1(Z; w) Received by the editors June 20, 1999 and, in revised form, June 15, 2001. 2000 Mathematics Subject Classification. Primary 19D10, 55P43. The second author was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 343, Bielefeld, Germany. c 2002 American Mathematical Society 3193 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3194 ROLAND SCHWANZL¨ AND ROSS STAFFELDT is a pushout diagram, or a generalized free product diagram, in the category of groups. Since the integral group ring functor is left adjoint to the functor which assigns to a ring with identity its group of invertible elements, passing to integral group rings yields another generalized free product diagram in the category of rings with identity. In case the maps leaving π1(W; w) are injective, all maps in the diagram are injective and the pushout is called a free product along an amalgamated subgroup. In [8], for example, the algebraic K-theory of the group rings arising in the injective situation is carefully studied. Generalizing the algebraic K-theory of rings, Waldhausen defined the algebraic K-theory of topological spaces in [10] and [9]. Applying the algebraic K-theory of spaces functor to diagram (1.1) yields a diagram: A(W ) /A(X) A(Y ) /A(Z) It is natural to ask how A(Z) is related to the other algebraic K-theories A(W ), A(X), A(Y ). In [1, page 125] one of Waldhausen's definitions for the algebraic K-theory of a topological space is reformulated as the algebraic K-theory of a \group ring" formed in the category of S-algebras. Namely, for a pointed connected simplicial set X,the algebraic K-theory of X may be defined as A(X)=K(S G(X) +), the K-theory of the S-algebra associated to the topological group G(X^j) whichj is the realization of the Kan loop group G(X)ofX. A definition of thej functorj S from pointed spaces to S-modules will be recalled below in Definition 2.4. Also,∧− according to [1], the category of S-algebras is cocomplete, so there exists a generalized free product (S G(X) +) S G(W ) + (S G(Y ) +), or pushout, associated to the three smaller spaces^j in thej diagram.∗ ^j j ^j j In[7]webeganthestudyofthealgebraicK-theory of generalized free products of S-algebras. If certain cofibration conditions on the S-algebra homomorphisms S G(W ) + S G(X) + and S G(W ) + S G(Y ) + are satisfied, then we can^j calculatej ! the K^j-theoryj of the generalized^j j free! product^j inj terms of the K-theory of the other three rings, plus a Nil-term. Indeed, the results follow the pattern of results obtained in [8] for the K-theory of group rings. In Theorem 1.1 below we prove that, if the diagram (1.1) is a pushout diagram of reduced simplicial sets, then an isomorphism of S-algebras (S G(X) +) S G(W ) + (S G(Y ) +) S G(Z) + ^j j ∗ ^j j ^j j −→ ^j j arises from the associated diagram of S-algebras by means of the universal property of the generalized free product. We can view this result as a stabilization and glob- alization of the classical fact recalled above, that a Seifert-van Kampen situation gives rise to a generalized free product diagram in the category of rings. It is a stabilization in the sense that the sphere spectrum S replaces the integers as the ring of coefficients, and it is a globalization in the sense that it involves the higher homotopy groups of the spaces in the diagram, as well as their fundamental groups. In particular, interesting examples are obtained by letting X and Y be cones on W ,sothatZ is the suspension of W . Moreover, we also show in Theorem 1.1 that the generalized free product diagram of S-algebras satisfies cellularity conditions that are stronger than the cofibration License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SEIFERT-VAN KAMPEN AND S-ALGEBRAS 3195 conditions we actually required in [7] to prove the K-theory decomposition results. Combining these results with the main result of [7] provides a decomposition the- orem for the algebraic K-theory of the space Z, stated as Corollary 1.2 below. The next section comprises the proof of Theorem 1.1. The proofs draw on some standard simplicial homotopy theory, some standard facts about the category of S-algebras, and some techniques developed in [7] for working on generalized free products of S-algebras. Our proof of the algebraic part of the result is essentially calculational; in turn, these calculations enable us to exhibit the cell structures we require for the application to K-theory. In section 3 we extend our first result to cover a more usual pushout diagram (1.3) W / /X Y / /Z of reduced simplicial sets in which only the horizontal arrows are cofibrations. We obtain in Theorem 3.1 that the diagram of S-algebras arising from this situation induces a homotopy equivalence of S-algebras (S G(X) +) S G(W ) + (S G(Y ) +) S G(Z) +: ^j j ∗ ^j j ^j j −→ ^j j Using this equivalence we explain how the K-theory fibration sequence of Corollary 1.2 generalizes to the more usual situation in Corollary 3.2. We conclude section 3 by explaining how to apply our results for pushout diagrams of reduced simplicial sets to pushout diagrams of connected pointed Kan sets and topological spaces. In the following statement we assume that each simplicial set is reduced, which means that the set of zero-simplices consists of a single element. In particular, a reduced simplicial set is connected. This hypothesis is technically convenient and involves no real loss of generality, as we explain at the end of section 3. Theorem 1.1. Let W / /X Y / /Z be a pushout diagram of reduced simplicial sets, where all simplicial sets are pointed by the zero-simplex w of W . Then the associated diagram of S-algebras S G(W ) + /S G(X) + ^j j ^j j S G(Y ) + /S G(Z) + ^j j ^j j induces an isomorphism (1.4) (S G(X) +) S G(W ) + (S G(Y ) +) S G(Z) + ^j j ∗ ^j j ^j j −→ ^j j from the generalized free product defined in [7] to the S-algebra S G(Z) +. All the S-algebras in the diagram above are cell S-algebras. Moreover,^j j the S- algebras S G(X) + and S G(Y ) + are cell S G(W ) +-modules. ^j j ^j j ^j j License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3196 ROLAND SCHWANZL¨ AND ROSS STAFFELDT In view of the cellularity properties of the maps to S G(X) + and S G(Y ) + and the isomorphism (1.4), Theorem 1.1 of ([7]) applies^j and therej is a fibration-up-^j j to-homotopy K( w;v) MVf ! K(S G(W ) +) K(S G(X) +) K(S G(Y ) +) K(S G(Z) +) ^j j × ^j j × ^j j ! ^j j where the fiber is the K-theory of the category of split Mayer-Vietoris presenta- tions of finite cell modules associated to the generalized free product diagram of S-algebras given in the theorem. If we make the appropriate substitutions, then we obtain the following corollary. Corollary 1.2. Given a diagram (1.1) there is a fibration-up-to-homotopy K( w;v) A(W ) A(X) A(Y ) A(Z): MVf −→ × × −→ 2.
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