At the end of his 1956 paper [On Manifolds Homeomorphic to the 7-Sphere](http://www.maths.ed.ac.uk/~aar/papers/exotic.pdf), 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](https://link.springer.com/book/10.1007%2Fb137100), 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?



Why is the first integral Pontryagin class a homeomorphism invariant?
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)](https://www.sciencedirect.com/science/article/pii/0166864185901002)


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)](https://arxiv.org/pdf/math/0105047.pdf))


