The intersecting kernels of Heegaard splittings

Lei, Fengchun¹ ; Wu, Jie²

¹School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
²Department of Mathematics, National University of Singapore, S17-06-02, 10 Lower Kent Ridge Road, Singapore 119076, Singapore

Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 887-908

Résumé

Let V ∪SW be a Heegaard splitting for a closed orientable 3–manifold M. The inclusion-induced homomorphisms π1(S) → π1(V ) and π1(S) → π1(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

where R = ℤ(π1(M)), J is a cyclic R–submodule of R{x1,…,xg}, Tϕ and θ are explicitly described morphisms of R–modules and Tϕ involves Fox derivatives related to the gluing data of the Heegaard splitting M = V ∪SW.

(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.

Détail

DOI : 10.2140/agt.2011.11.887

Keywords: Heegaard splitting, intersecting kernel, $3$–manifold, mapping class, Riemann surface

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