The intersecting kernels of Heegaard splittings

Let $V \cup_{S} W$ be a Heegaard splitting for a closed orientable $3$ –manifold $M$ . The inclusion-induced homomorphisms $π_{1} (S) \to π_{1} (V)$ and $π_{1} (S) \to π_{1} (W)$ are both surjective. The paper is principally concerned with the kernels $K = Ker (π_{1} (S) \to π_{1} (V))$ , $L = Ker (π_{1} (S) \to π_{1} (W))$ , their intersection $K \cap L$ and the quotient $(K \cap L) ∕ [K, L]$ . The module $(K \cap L) ∕ [K, L]$ is of special interest because it is isomorphic to the second homotopy module $π_{2} (M)$ . There are two main results.

(1) We present an exact sequence of $ℤ (π_{1} (M))$ –modules of the form

(K \cap L) ∕ [K, L] ↪ R {x_{1}, \dots, x_{g}} ∕ J \overset{T^{ϕ}}{\to} R {y_{1}, \dots, y_{g}} \overset{θ}{\to} R \overset{ϵ}{↠} ℤ,

where $R = ℤ (π_{1} (M))$ , $J$ is a cyclic $R$ –submodule of $R {x_{1}, \dots, x_{g}}$ , $T^{ϕ}$ and $θ$ are explicitly described morphisms of $R$ –modules and $T^{ϕ}$ involves Fox derivatives related to the gluing data of the Heegaard splitting $M = V \cup_{S} W$ .

(2) Let $K$ be the intersection kernel for a Heegaard splitting of a connected sum, and $K_{1}$ , $K_{2}$ the intersection kernels of the two summands. We show that there is a surjection $K \to K_{1} * K_{2}$ onto the free product with kernel being normally generated by a single geometrically described element.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57M27, 57M99, 20F38

Secondary: 57M05, 37E30

References

Publication

Received: 25 July 2010

Revised: 29 December 2010

Accepted: 12 January 2011

Published: 25 March 2011

Authors

Fengchun Lei
School of Mathematical Sciences Dalian University of Technology Dalian 116024 China

Jie Wu
Department of Mathematics National University of Singapore S17-06-02, 10 Lower Kent Ridge Road Singapore 119076 Singapore
http://www.math.nus.edu.sg/~matwujie

Volume 11, issue 2 (2011)

Fengchun Lei and Jie Wu

Abstract

Keywords

Mathematical Subject Classification 2000

References

Publication

Authors