At the end of his 1956 paper On Manifolds Homeomorphic to the 7-Sphere, Milnor shows that either

  1. There exists a closed topological 8-manifold with no smooth structure; or
  2. The first Pontryagin class $p_1$ of an open smooth 8-manifold is not a topological invariant.

We now know that 1 is true.

In section 4.4 of The Novikov Conjecture - Geometry and Algebra, Kreck and Lück outline a proof that $p_i$ is not a homeomorphism invariant for $i>1$ and claim that $p_1$ is a homeomorphism invariant but do not supply a proof. They reference a manuscript (On the topological invariance of Pontrjagin classes) of Kreck and say that it is in preparation, but said manuscript does not seem to have been published since.

Does anyone have a reference or proof for $p_1$ being a topological invariant?

  • 1
    $\begingroup$ (Thank you, I did that just now.) Even if Milnor formulates Theorem 4 for the integer class $p_1$, the given proof only uses the rational one (so he proved more then formulated). Because of this, it is fair to say that invariance of the integer $p_1$, however interesting by itself, is irrelevant to this paper of Milnor. (The rational Pontryagin classes were shown to be topologically invariant by Novikov in 1966.) $\endgroup$ – Alex Gavrilov Feb 3 '18 at 14:01

The topological invariance of the first Pontryagin class is proved in the paper

  • B.L. Sharma. Topologically invariant integral characteristic classes. Topology Appl. 21 (1985), no. 2, 135–146. (link to Elsevier website)

In low degrees, Sharma computed the smallest multiples of the Pontryagin classes which are topological invariants, and Theorem 1.6 of his paper states that that multiple is 1 for the first Pontryagin class.

(Note: I found that via a discussion Section 22 of Rudyak's "Piecewise linear structures on manifolds", (arxiv link))

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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.

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