Vanishing of whitehead torsion in dimension four

Abstract

IN TOPOLOGY and geometry it is often useful or important to recognize n-dimensional manifolds that are isomorphic to products X x [0, l] of a compact manifold X with the unit interval. The s-cobordism theorem of Barden, Mazur and Stallings (see [13, 171) states that a compact (n + 1)-manifold W of dimension n + 1 2 6 with boundary 8 W = M,, u Ml is a product M0 x [0, l] if and only if W is an h-cobordism (i.e., W is homotopically a product) and a certain algebraic invariant z( W; M,,) called the Whitehead torsion is trivial (for n = 4 the results of [8,9] yield a topological version of the s-cobordism theorem for a large class of the fundamental groups). The Whitehead torsion invariant takes values in an abelian group called the Whitehead group of M that depends only on the fundamental group and is denoted by Wh(n,( W)). The vanishing condition on Z( W; M,-,) is essential because the Whitehead group is nonzero in many cases and every element can be realized as the torsion of some h-cobordism ( W”+l; M”,, My) for n 2 4; a proof in the case n 2 5 appears in [13], and the case n = 4 is treated in [2]. In fact, it is possible to choose the h-cobordisms so that M,-, ?z Ml (compare [2, Prop. 3.3 and the first sentence in the paragraph on p. 515 before Prop. 3.21). On the other hand, our understanding of the case n = 3 is still quite limited. For example, the following realization question from [16, Problem 4.91 is still open.