On Incidence Matrices, Nuclei and Homotopy Types

The main object of this paper is to establish as close a connection as possible between certain purely algebraic processes on the one hand, and the theory of nuclei' and homotopy types2 of complexes on the other. The algebraic processes include the transformations, operating on the incidence matrices with elements in the group ring of a given complex, by means of which Reidemeister's invariants are defined.3 But in the main theorems, namely theorems 8 and 9 in ?4, the incidence matrices r', ... , rl are replaced by what we call a system for a given complex KV. A system, (r, R), consists of the matrices rn, *-, r3, together with a natural system, R, of generators and relations, for the fundamental group 7r,(Kn). Such systems, defined purely algebraically, are classified by two kinds of equivalence, called L-equivalence and L*-equivalence, of which the former implies the latter. Theorem 8 asserts two kinds of combinatorial invariance. It states that systems for two given complexes K n and K' are L-equivalent if K' and K n have the same nucleus, and L*-equivalent if K' and K1 are of the same homotopy type. Theorem 9 is complementary to theorem 8, and states that the elementary transformations by means of which X-equivalence is defined (X = L or L*), can be copied geometrically. More precisely, if (r, R) is a system for K , then any system which is X-equivalent to (r, R), is a system for some complex of the same homotopy type as Kn, and for one which has the same nucleus as Kn if X = L. This means that one can carry out certain types of algebraic calculation without destroying the geometrical significance of the system on which one is operating, and the result is a step towards translating the theory of nuclei and m-groups, likewise homotopy types (of polyhedra), into purely algebraic terms. As an application of the theorems in ??3 and 4 we prove, in ?5, that two lens