Killing characteristic classes by surgery

Abstract

Let M be an n-dimensional C ∞ {C^\infty } manifold and let \[ c ∈ H ∗ ( B O ; Z 2 ) c \in {H^\ast }(BO;{Z_2}) \] be a characteristic class. Suppose that c as a factor annihilates all the characteristic numbers of M. We prove that if dim ⁡ c ⩾ ( n + 1 ) / 2 \dim c \geqslant (n + 1)/2 then M is cobordant to a manifold which has the class c zero, in that way answering in the affirmative a question raised by C. T. C. Wall. We examine the same question for more general cobordism theories, and for Z p {Z_p} characteristic classes.