TITLE: Relations among Pontryagin numbers
QUESTION [15 upvotes]: The Hattori-Stong theorem describes the image of the morphism:
$$\tau:\Omega^{SO}_*\rightarrow H_*(BSO;\mathbb{Q})$$
that associates to any closed, smooth, oriented manifold $M$ the homology class $\phi_*([M])$ where $\phi$ is the classifying map $M\stackrel{\phi}{\rightarrow}BSO$ of the tangent bundle and $[M]$ is the fundamental class of $M$.
This theorem describes all the relations satisfied by the Pontryagin numbers of smooth manifolds in terms of integrality conditions.
Question: do we know the topological analogue of the Hattori-Stong theorem, i.e. do we know all the relations that Pontryagin numbers of topological manifolds should satisfy? 
Edit: as we have an isomorphism $\Omega^{Top}_*/Tors\cong \Omega^{PL}/Tors$, this question also amounts to knowing all the relations that Pontryagin numbers of $PL$-manifolds should satisfy.

REPLY [2 votes]: Check this out: Madsen-Milgram, The Classifying Spaces for Surgery and Cobordism of Manifolds, Corollary 11.26.