Representing Handlebodies by Plumbing and Surgery

Abstract

{/ 6 and n ^>2, JF is diffeomorphic to the sum of (TTI—- n,)-disk bundles over ^-spheres. This is obtained by constructing disjoint spheres S* each of which has D' = D X o as the one hemisphere and a disk D in Dm as the other hemisphere and by tying with bands the tubular neighbourhoods of the zi-sheres Sn{ = Dt}\UDjn. But, if 27i>77i, there arise non-empty intersections of the 7i-disks D. In this paper, we consider how handlebodies can be constructed from (m — ^)-disk bundles over n -spheres in the case when 2n'>m. This problem was motivated to analyze the structure of such m -manifolds with non-trivial homology groups only at dimensions 0, k, m — k, and m, (m^ 2k). The results are given in the section 6 as an application. In Hi 3], J. Milnor constructed the manifolds which are parallelizable and have exotic spheres as their boundaries. There, some 2A;-disk bundles over 2&-spheres are plumbed correspondingly with a suitable matrix, and in order to kill the fundamental group of the plumbing manifold a proper disk is attached to the boundary. In CllH, M. Kerevaire also constructed a manifold like that by plumbing 5-disk bundles over 5-spheres. In this paper, we generalize those constructions of J. Milnor and M. Kervaire. The generalization is done by plumbing (m — ra)-disk bundles over n,-spheres along (2n — m)-spheres or tori which are imbedded in the ^-spheres,