Let be a Heegaard splitting for a closed orientable –manifold . The inclusion-induced homomorphisms and are both surjective. The paper is principally concerned with the kernels , , their intersection and the quotient . The module is of special interest because it is isomorphic to the second homotopy module . There are two main results.
(1) We present an exact sequence of –modules of the form
where , is a cyclic –submodule of , and are explicitly described morphisms of –modules and involves Fox derivatives related to the gluing data of the Heegaard splitting .
(2) Let be the intersection kernel for a Heegaard splitting of a connected sum, and , the intersection kernels of the two summands. We show that there is a surjection onto the free product with kernel being normally generated by a single geometrically described element.
"The intersecting kernels of Heegaard splittings." Algebr. Geom. Topol. 11 (2) 887 - 908, 2011. https://doi.org/10.2140/agt.2011.11.887