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2011 The intersecting kernels of Heegaard splittings
Fengchun Lei, Jie Wu
Algebr. Geom. Topol. 11(2): 887-908 (2011). DOI: 10.2140/agt.2011.11.887

Abstract

Let VSW 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 KL and the quotient (KL)[K,L]. The module (KL)[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 { x 1 , , x g } J T ϕ R { y 1 , , y g } θ R ϵ ,

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=VSW.

(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 KK1K2 onto the free product with kernel being normally generated by a single geometrically described element.

Citation

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Fengchun Lei. Jie Wu. "The intersecting kernels of Heegaard splittings." Algebr. Geom. Topol. 11 (2) 887 - 908, 2011. https://doi.org/10.2140/agt.2011.11.887

Information

Received: 25 July 2010; Revised: 29 December 2010; Accepted: 12 January 2011; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1215.57007
MathSciNet: MR2782546
Digital Object Identifier: 10.2140/agt.2011.11.887

Subjects:
Primary: 20F38, 57M27, 57M99
Secondary: 37E30, 57M05

Rights: Copyright © 2011 Mathematical Sciences Publishers

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