Journal of Pure and Applied Algebra 77 (1992) 39-65 North-Holland 39 Mayer-Vietoris sequences in homotopy of 2-complexes and in homology of groups William A. Bogley and Mauricio A. Gutikrez zyxwvutsrqponmlkjihgfedcba Depurtment of Mathematics, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED Tufts University, Medford, MA 021.%5, USA Communicated by K.W. Gruenberg Received 24 April 1990 Revised 20 November 1990 Introduction This paper is concerned with the second homotopy groups (both absolute and relative) of a union of CW complexes, and with the integral homology of groups in a pushout square. For the last of these, a simple and useful version of our result is as follows. Theorem. Let R and S be normal subgroups of a free group F. There is a long exact sequence (A) H,FIR@ H,FIS+ H,FIRS* R n S n [F, F] IR, Sl[F, R n Sl + H,FIR@ H,FIS+ H,FIRS -+ (R n S:[F, F] - H,FIR@ H,FlS+ H,FIRS*O involving integral homology groups. q This sequence is natural in an appropriate sense, and is a useful tool for homology calculations involving multi-relator groups. It also yields a relative Hopf formula for third homology (Corollary 5.5). One ingredient in the proof of the theorem is due to Hopf [9, 101: If X is a two-complex with fundamental group G, then there is an exact sequence (B) O+ H,G* ZgG rr,X* H,X+ H,G+O which is natural in maps of two-complexes. Suppose next that a connected CW 0022.4049/921$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved