Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 11, Number 2 (2011), 887-908.
The intersecting kernels of Heegaard splittings
Let be a Heegaard splitting for a closed orientable –manifold . The inclusion-induced homomorphisms and are both surjective. The paper is principally concerned with the kernels , , their intersection and the quotient . The module is of special interest because it is isomorphic to the second homotopy module . There are two main results.
(1) We present an exact sequence of –modules of the form
where , is a cyclic –submodule of , and are explicitly described morphisms of –modules and involves Fox derivatives related to the gluing data of the Heegaard splitting .
(2) Let be the intersection kernel for a Heegaard splitting of a connected sum, and , the intersection kernels of the two summands. We show that there is a surjection onto the free product with kernel being normally generated by a single geometrically described element.
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
Permanent link to this document
Digital Object Identifier
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
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