Fibering Five Manifolds Over A Circle.
Chang-wang Felix Hsu · Deep Blue (University of Michigan) · 1985
In this thesis, after we update some basic facts about 4- manifolds and the smoothings of 4 dimensional manifolds we prove the following: Technical Lemma. Given a 5-dimensional compact cobordism (W;V(,1),V(,2)) satisfying: (a) Int(W) is smooth, (b) W, V(,1), V(,2) are 1-connected, (c) H(,2)(W,V(,1)) = , H(,2)(W,V(,2)) = are free abelian of rank j, k respectively, and (d) H(,3)(W) = 0 and H(,2)(W) is free abelin; then (1) j = k, (2) V(,1) = V(,0) # k(S('2) x S('2)) where V(,0) is a closed, 1-connected topological 4-manifold. Similarly V(,2) is the sum of a closed 1-connected manifold and k copies of (S('2) x S('2)). (3) Each generator of H(,3)(W,V(,1)) = can be represented as a 3-handle. We then use this lemma to deduce the following fibering theorem: Theorem. If W is a smooth closed 5-manifold with (pi)(,1)(W) = , and H(,*)((')W) is finitely-generated, then W fibers over a circle with fiber a 1-connected topological 4-manifold. As a second application, we use our technical lemma to provide a unified proof for some well-known theorems in 5-dimensional knot theory. We conclude with an interesting corollary which should shed some light on the classification of closed, oriented smooth 5-manifolds with (pi)(,1) = . Corollary. The closed, connected, orientable 5-manifolds M with (pi)(,1)(M) = and H(,2)(M) (TBOND) 0 are completely classified by their Milnor forms, hence they are classified by their Blanchfield forms also.