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The intersecting kernels of Heegaard splittings

Fengchun Lei and Jie Wu

Algebraic & Geometric Topology 11 (2011) 887–908

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 (π1(M))–modules of the form

(K L)[K,L]R{x1,,xg}JTϕR{y 1,,yg}θRϵ,

where R = (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 KK1 K2 onto the free product with kernel being normally generated by a single geometrically described element.

Heegaard splitting, intersecting kernel, $3$–manifold, mapping class, Riemann surface
Mathematical Subject Classification 2000
Primary: 57M27, 57M99, 20F38
Secondary: 57M05, 37E30
Received: 25 July 2010
Revised: 29 December 2010
Accepted: 12 January 2011
Published: 25 March 2011
Fengchun Lei
School of Mathematical Sciences
Dalian University of Technology
Dalian 116024
Jie Wu
Department of Mathematics
National University of Singapore
S17-06-02, 10 Lower Kent Ridge Road
Singapore 119076