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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.

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

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    $\begingroup$ Could you please state that corollary for the benefit of those who do not have easy access to this work? $\endgroup$
    – Todd Trimble
    Commented Mar 3, 2017 at 13:24
  • $\begingroup$ @Todd Trimble: sorry. the description of the image is not so easy to state, but the paper is the first hit on google $\endgroup$
    – nikola karabatic
    Commented Mar 3, 2017 at 21:33
  • $\begingroup$ For the convenience of readers: maths.ed.ac.uk/~aar/papers/madmil.pdf, where Corollary 11.26 is titled the PL Hattori-Stong Theorem. $\endgroup$
    – Todd Trimble
    Commented Mar 5, 2017 at 21:09

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