Consider the following problem: What is the relation between the group of oriented cobordism classes of oriented Poincare duality spaces (denoted QPD.) and the stable homotopy group wrMSG, where MSG is the Thom space which arises from the universal orientable spherical fibrations. In particular, the result of Spivak [8] allows the construction of a Pontrjagin-Thom map p: QPD.wr MSG. One therefore, wishes to know how close p is to being an isomorphism. Of course, it is not true that p is an isomorphism, for it is easily seen that r*MSG is a finite group in each dimension whereas, in dimensions 4k, the index of an oriented Poincare duality space defines a non-trivial homomorphism Q'-'D Z. Or, to put it another way, there exist two Poincare duality spaces M4k and N4k, with a degree one map f: M4k N4k covered by a map of Spivak normal bundles, (which in turn extends to a degree-one map of ambient spheres S4k+m S4k+m m large) yet with index M# index N. For example, the non-smoothable Milnor manifolds of dimension 4k have fiberhomotopy trivial normal bundles, (i.e. trivial normal fibrations when regarded as P.D. spaces) yet none of them have zero index, hence none is oriented cobordant to zero.