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Mayer-Vietoris sequences in homotopy of 2-complexes and in homology of groups. (English) Zbl 0761.55016

The Mayer-Vietoris sequence deals with the situation of a CW-complex \(K=K_ 1\cup K_ 2\), and determines invariants of \(K\) in terms of those of \(K_ 1,K_ 2,K_ 0=K_ 1\cap K_ 2\) and morphisms induced by inclusion. For second relative homotopy groups, a 2-dimensional version of the van Kampen Theorem, due to the reviewer and P. J. Higgins [Proc. Lond. Math. Soc., III. Ser. 36, 192-212 (1978; Zbl 0405.55015)], implies that the natural morphism \(\sigma:\pi_ 2(K_ 1,K_ 0)\circ\pi_ 2(K_ 2,K_ 0)\to\pi_ 2(K,K_ 0)\) is an isomorphism of crossed \(\pi_ 1K_ 0\)-modules, if all the spaces are connected, and \((K_ 1,K_ 0)\), \((K_ 2,K_ 0)\) are 1-connected. Here \(\circ\) denotes the coproduct of these crossed \(\pi_ 1K_ 0\)-modules. This paper uses this result, and deals with related situations. For example if \(R\), \(S\) are normal subgroups of a free group \(F\), an eight term exact homology sequence is obtained involving the low dimensional homologies of \(F/R\), \(F/S\), \(F/RS\).
A consequence is that if \(F/R\), \(F/S\) are also free, then \(H_ 3(F/RS)\) is isomorphic to the group \((R\cap S\cap[F,F]/([R,S][F,R\cap S])\). This recovers the case \(n=3\) of the generalized Hopf formula for \(H_ n(G)\) due to the reviewer and G. J. Ellis [Bull. Lond. Math. Soc. 20, 124-128 (1988; Zbl 0611.20032)]. Another result gives conditions under which the above map is an isomorphism in the case when \(K\) is a 2- complex. Methods here include results on projective crossed modules, due to M. Dyer [Combinatorial group theory and topology, Ann. Math. Stud. 111, 255-264 (1987; Zbl 0643.57005)]. A number of group-theoretic examples are given illustrating the results.
Reviewer: R.Brown

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U05 Abstract complexes in algebraic topology
20J05 Homological methods in group theory
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[1] Brandenburg, J.; Dyer, M.; Strebel, R., On Whitehead’s aspherical question, II, Contemp. Math., 20, 65-78 (1983) · Zbl 0531.57004
[2] Brown, R., Coproducts of crossed \(P\)-modules: Applications to second homotopy groups and to the homology of groups, Topology, 23, 337-345 (1984) · Zbl 0519.55009
[3] Brown, R.; Ellis, G., Hopf formulae for the higher homology of a group, Bull. London Math. Soc., 20, 124-128 (1988) · Zbl 0611.20032
[4] Brown, R.; Higgins, P., On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc., 36, 193-212 (1978), (3) · Zbl 0405.55015
[5] Dyer, M., Subcomplexes of two-complexes and projective crossed modules, (Gersten, S.; Stallings, J., Combinatorial Group Theory and Topology. Combinatorial Group Theory and Topology, Annals of Mathematics Studies, 111 (1987), Princeton University Press: Princeton University Press Princeton, NJ), 255-264
[6] Fenn, R., Techniques in Geometric Topology, London Mathematical Society Lecture Note Series, 57 (1983), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0517.57001
[7] Gersten, S., Reducible diagrams and equations over groups, (Essays in Group Theory. Essays in Group Theory, Mat. Sci. Res. Inst., 8 (1987), Springer: Springer Berlin), 15-74 · Zbl 0644.20024
[8] Gutiérrez, M.; Ratcliffe, J., On the second homotopy group, Quart. J. Math. Oxford, 32, 45-55 (1981), (2) · Zbl 0457.57002
[9] Hopf, H., Fundamentalgruppe und zweite Bettische Gruppe, Comment Math. Helv., 14, 257-309 (1941) · JFM 68.0503.01
[10] Hopf, H., Beitrage zur Homotopietheorie, Comment Math. Helv., 17, 307-326 (1945) · Zbl 0061.40704
[11] Huebschmann, J., Cohomology theory of aspherical groups and of small cancellation groups, J. Pure Appl. Algebra, 14, 137-143 (1979) · Zbl 0396.20021
[12] Johnson, D.; Robertson, E., Finite groups of deficiency zero, (Wall, C., Homological Group Theory. Homological Group Theory, London Mathematical Society Lecture Note Series, 36 (1979), Cambridge University Press: Cambridge University Press Cambridge), 275-289 · Zbl 0423.20029
[13] Ladra, M.; Rodriguez, A., Sobre un resultado de R. Brown, (Actas X Jornadas Hispano-Lusas de Matematicas (1985), University of Murcia: University of Murcia Spain), 25-28
[14] Lyndon, R., Cohomology theory of groups with a single defining relation, Ann. of Math., 52, 650-665 (1950) · Zbl 0039.02302
[15] Pride, S., The relation module structure of weakly non-spherical groups, Preprint (1989), University of Glasgow
[16] Pride, S., Identities among relations of group presentations, (Ghys, E.; Haefliger, A.; Verjovsky, A., Proceedings of the Workshop on Group Theory from a Geometrical Viewpoint, International Centre of Theoretical Physics, Trieste (1990), World Scientific Publishing: World Scientific Publishing Singapore), to appear · Zbl 0843.20026
[17] Pride, S.; Stohr, R., The (co)-homology of aspherical Coxeter groups, J. London Math. Soc., 42, 49-63 (1990), (2) · Zbl 0727.20036
[18] Sieradski, A., A combinatorial interpretation of the third homology of a group, J. Pure Appl. Algebra, 33, 81-96 (1984) · Zbl 0551.20039
[19] Stammbach, U., Homology in Group Theory, Lecture Notes in Mathematics, 359 (1973), Springer: Springer Berlin · Zbl 0272.20049
[20] Whitehead, J., On adding relations to homotopy groups, Ann. of Math., 47, 806-810 (1946) · Zbl 0060.41104
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