Algebraic & Geometric Topology

The intersecting kernels of Heegaard splittings

Fengchun Lei and Jie Wu

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57M99: None of the above, but in this section 20F38: Other groups related to topology or analysis
Secondary: 57M05: Fundamental group, presentations, free differential calculus 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces

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


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.

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