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Indexed by:期刊论文

Date of Publication:2011-01-01

Journal:ALGEBRAIC AND GEOMETRIC TOPOLOGY

Included Journals:Scopus、SCIE

Volume:11

Issue:2

Page Number:887-908

ISSN No.:1472-2739

Abstract:Let V boolean OR(S) W be a Heegaard splitting for a closed orientable 3-manifold M. The inclusion-induced homomorphisms pi(1)(S) -> pi(1)(V) and pi(1)(S) -> pi(1)(W) are both surjective. The paper is principally concerned with the kernels K = Ker(pi(1)(S) -> pi(1)(V)), L = Ker(pi(1)(S) -> pi(1)(W)), their intersection K boolean AND L and the quotient (K boolean AND L) / [K, L]. The module (K boolean AND L) / [K, L] is of special interest because it is isomorphic to the second homotopy module pi(2)(M). There are two main results.
   (1) We present an exact sequence of Z(pi(1)(M))-modules of the form
   (K boolean AND L)/[K, L] curved right arrow R{x(1,) ...., x(g)}/J -> T phi R{y1, ..., yg}-> theta R -> epsilon Z,
   where R = Z(pi(1)(M)), J is a cyclic R-submodule of R{x(1,) ..., x(g)}, T(phi) and theta are explicitly described morphisms of R-modules and T(phi) involves Fox derivatives related to the gluing data of the Heegaard splitting M = V boolean OR(S) W. () Let kappa(2) be the intersection kernel for a Heegaard splitting of a connected sum, and kappa(1), kappa(2) the intersection kernels of the two summands. We show that there is a surjection kappa -> kappa(1) * kappa(2) onto the free product with kernel being normally generated by a single geometrically described element.