Orientability of manifolds for generalised homology theories

Abstract

Recently, generalised homology theories have received much attention ([5], [6], [19] etc.). An interesting question which arises in these theories is the orientability of manifolds [20]; as Whitehead remarks, this is a delicate question. In this paper we shall show that if there exists a microbundle 4 over a closed manifold M such that 4 is A-orientable in the sense of Dold [6], and such that the top homology class of the Thom space M4 is spherical, then M is A-orientable in the sense of Whitehead [19]. Hence M satisfies Poincare duality with coefficient spectrum A. Let N be a compact smooth manifold whose boundary ON= Y is a homotopy sphere which bounds a 7r-manifold: form a manifold M by adjoining to N a cone on its boundary S. We shall call such manifold-s M almost-smooth: the manifolds of Kervaire [11], Smale [16], Eells and Kuiper [8] (see also [18]), which do not have the same homotopy type as any closed smooth manifold, are all almost smooth. We shall show that, over any such M, there is a vector bundle 4 such that the top homology class of its Thom complex MX is spherical. It follows that (provided M is orientable in the usual sense), M is orientable for K-theory and KO-theory [2] and for bordism theory [1]. It also follows from the existence of such bundles that the hypothesis on the signature in the theorem of Browder [4], which gives necessary and sufficient conditions for a finite, simply-connected CW-complex to have the homotopy type of a closed smooth manifold (of dimension : 3,4), cannot be dispensed with. Microbundles and Thom spaces. We consider pairs (i,j) of maps, with ji the identity map of some fixed space B. Two pairs (i1,jl) and (i2,j2) are equivalent if there is a third pair (i3,j3) and a commutative diagram,