Halving surfaces in 3-manifolds (Q1823513)

An orientable surface F halves a connected orientable closed 3-manifold M if M-F consists of two connected components with diffeomorphic closures. This generalizes the concept of a Heegaard decomposition where the components of M-F are handlebodies. The halving number h(M) is the minimal integer g for which M can be halved by a closed surface of genus g. The aim is to find a lower bound of h(M) in terms of cohomological invariants of M. Let H*(M) be the cohomology ring of M with coefficients in some field. Let d(M) denote the greatest integer d for which there are linear subspaces \(V_ 1\), \(V_ 2\) of dimension d in \(H^ 1(M)\) such that \(V_ 1\cap V_ 2=\emptyset\) and the cup product \(e_ 1\cup e_ 2=0\) for any \(e_ 1\in V_ 1\), \(e_ 2\in V_ 2\). Then h(M)\(\geq \dim H^ 1(M)-2d.\) This is proved using standard arguments of homology theory, the Poincaré duality and the fact that for an odd dimensional manifold with boundary the Euler characteristic of the boundary is twice the Euler characteristic of the manifold. Moreover it is shown: M is a homotopy sphere if \(h(M)=0\) and \(\pi_ 2(M)=0\). If M is a fibre bundle over a surface then \(0\leq h(M)\leq 3\). If M is ``halved'' by an orientable surface consisting of c components of genera \(g_ 1,...,g_ c\) then \(h(M)\leq (c-1)- \sum^{\infty}_{i=1}g_ i.\)