Amalgamations and the Kervaire Problem

Following S. Brick, a 2-complex X is called "Kervaire" if all systems of equations, with coefficients in arbitrary groups G and the attaching maps of X as the words in the variable letters, are solvable in an over group of G. An obstruction theory is developed for solving equations modeled on Z = Xp Y, where X and Y are Kervaire 2-complexes and T is a subgraph of Z^\ each connected component of which injects at the 7Ti-level into iri(Z). A 2-complex of the form K(x, y\ w(x) = w'(y)) is Kervaire, where w{x) and w'{y) are (not necessarily reduced) words which do not freely reduce to 1. The Kervaire problem [7, p. 403] originally asked whether a nontrivial group can be killed by adjoining a single free generator and a single relator. This problem has been vastly generalized by Howie [5], who asked whether a system of equations over an arbitrary coefficient group G, whose words in the variable letters are the attaching maps of a 2-complex X with H2{X) — 0, is solvable in an overgroup of G. It is convenient to introduce a terminology due to S. Brick [1] who calls a 2-complex X Kervaire iff all systems of equations over all coefficient groups G modeled on the attaching maps of X are solvable in an overgroup of G. Thus, e.g., the dunce hat K(x\xxx) is Kervaire because Howie has shown that the equation axbxcx = 1, with a, 6, c G G, can always be solved in an overgroup of G [6]. In this terminology, a nontrivial group can never be killed by adjoining a single free generator and a single relator iff the 2-complex K(x\w(x)) is Kervaire, where w(x) is a word in x and x~ whose exponent sum in x is dbl. For a 2-complex with one 2-cell X = K(x\, £2, • • •, xn\w(x)) Howie's problem can be shown (nontrivially) to imply that X is Kervaire iff w(x) does not freely reduce to 1 (the "if" assertion is the nontrivial one here). Since X = K(x\w(x)) can be easily shown to be Cockcroft iff w(x) does not freely reduce to 1, Howie's problem for 2-complexes X with one 2-cell amounts to Received by the editors September 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 20F05, 57M20.