For spin manifolds this is proved in Corollary 1.22 (p.17) of Kammeyer's Diploma. Kreck's claim that the "spin" assumption can be dropped is mentioned after the corollary, with the caveat that "the author was unable to locate such a paper".

For spin manifolds this is proved in Corollary 1.22 (p.17) of Kammeyer's Diploma. Kreck's claim that the "spin" assumption can be dropped is mentioned after the corollary, with the caveat that "the author was unable to locate such a paper".

Incidentally, in dimension 4 the Pontryagin class (of a closed oriented smooth manifold) is proportional to the signature, and so is a homotopy invariant.