The intersecting kernels of Heegaard splittings. (English) Zbl 1215.57007

Let \(M=V\cup_{S}W\) be a Heegaard splitting, where \(V\) and \(W\) are handlebodies of genus \(n\) and \(S=\partial V=\partial W.\) The authors investigate the (possibly singular) curves on the Riemann surface \(S\) that can be extended to (possibly singular) disks in \(V\) and \(W,\) respectively. The inclusion-induced homomorphisms \(\pi_{1}(S)\rightarrow\pi_{1}(V)\) and \(\pi_{1}(S)\rightarrow \pi_{1}(W)\) are both surjective. The paper is principally concerned with the kernels \(K=\text{Ker}(\pi_{1}(S)\rightarrow\pi_{1}(V)),\) \(L=\text{Ker}(\pi_{1} (S)\rightarrow\pi_{1}(W)),\) 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 \(\pi_{2}(M).\) In the paper there are two main results. 6.5mm
The authors present an exact sequence of \(\mathbb{Z}(\pi_{1} (M))\)-modules of the form \((K\cap L)/[K,L]\hookrightarrow R\{x_{1} ,\dots,x_{g}\}/J\overset{T^{\phi}}{\rightarrow}R\{y_{1},\dots,y_{g}\}\overset {\theta}{\rightarrow}R\twoheadrightarrow\mathbb{Z},\) where \(R=\mathbb{Z} (\pi_{1}(M)),\) \(J\) is a cyclic \(R\)-submodule of \(R\{x_{1},\dots,x_{g}\},\) \(T^{\phi}\) and \(\theta\) are explicitly described morphisms of \(R\)-modules and \(T^{\phi}\) involves Fox derivatives related to the gluing data of the Heegaard splitting \(M=V\cup_{S}W.\)
Let \(\mathcal{K}\) be the intersection kernel for a Heegaard splitting of a connected sum, and \(\mathcal{K}_{1},\) \(\mathcal{K}_{2}\) the intersection kernels of the two summands. The authors show that there is a surjection \(\mathcal{K\rightarrow K}_{1}\ast\mathcal{K}_{2}\) onto the free product with kernel being normally generated by a single geometrically described element.


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M99 General low-dimensional topology
20F38 Other groups related to topology or analysis
57M05 Fundamental group, presentations, free differential calculus
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
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