The intersecting kernels of heegaard splittings

Abstract

Let V ∪sW be a Heegaard splitting for a closed orientable 3-manifold M. The inclusion-induced homomorphisms π1 (S) → π 1 (S)→ π 1 (W) are both surjective. The paper is principally concerned with the kernels K = Ker (π1(S) → π1 (V)), L = Ker(π1(S) → π1 (W)), their intersection K ∩ L and the quotient (K∩L)/[K,L]. The module (K ∩ L)/=[K,L] is of special interest because it is isomorphic to the second homotopy module π2(M). There are two main results. (1) We present an exact sequence of Z (π1(M)-modules of the form (K∩L)/=[K,L] {right arrow, hooked} R{xl,...,xg}/J →Tπ R{y1,...yg}→theta; R↠∈Z where R=Z(φ1(M)), J is a cyclic R-submodule of R{X1...xg}, Tθ and are explicitly described morphisms of R-modules and T involves Fox derivatives related to the gluing data of the Heegaard splitting M = V Us W. (2) Let K be the intersection kernel for a Heegaard splitting of a connected sum, and k1, K2 the intersection kernels of the two summands. We show that there is a surjection K → K1 * K2 onto the free product with kernel being normally generated by a single geometrically described element.