## Algebraic & Geometric Topology

### 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 $π1(S)→π1(V)$ 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

$( K ∩ L ) ∕ [ K , L ] ↪ R { x 1 , … , x g } ∕ J → T ϕ R { y 1 , … , y g } → θ R ↠ ϵ ℤ ,$

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 2 (2011), 887-908.

Dates
Revised: 29 December 2010
Accepted: 12 January 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715212

Digital Object Identifier
doi:10.2140/agt.2011.11.887

Mathematical Reviews number (MathSciNet)
MR2782546

Zentralblatt MATH identifier
1215.57007

#### Citation

Lei, Fengchun; Wu, Jie. The intersecting kernels of Heegaard splittings. Algebr. Geom. Topol. 11 (2011), no. 2, 887--908. doi:10.2140/agt.2011.11.887. https://projecteuclid.org/euclid.agt/1513715212

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