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The Wikipedia page for Rokhlin's Theorem says

"Michael Freedman's E8 manifold is a simply connected compact topological manifold with vanishing $w_{2}(M)$ and intersection form $E_{8}$ of signature 8. Rokhlin's theorem implies that this manifold has no smooth structure."

Is it easy to see that $w_2(M) = 0$?

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    $\begingroup$ Wu's formula tells you that on a simply-connected $4$-manifold $M$, $w_2(M)$ is characteristic, i.e. $\langle x,x\rangle = \langle w_2(M),x\rangle \pmod 2$. In particular, $w_2(M)$ vanishes iff the intersection form is even, which $E_8$ is. $\endgroup$
    – Bertram Arnold
    Commented May 30, 2021 at 8:30
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    $\begingroup$ How does one define $w_2$ of a topological manifold? Does Wu's formula still hold? (Not that one really needs it for this argument to work, but as phrased it seems relevant.) $\endgroup$
    – Marco Golla
    Commented May 30, 2021 at 18:57
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    $\begingroup$ The Stiefel-Whitney classes lift from $BO(n)$ to $BTop(n)$, the classifying space of microbundles, essentially by the definition as the failure of the Thom isomorphism to commute with Steenrod operations. Every topological bundle has a canonical microbundle, which is a lift of the tangent bundle of the manifold is equipped with a smooth structure. See also this question: mathoverflow.net/questions/243629/… $\endgroup$
    – Bertram Arnold
    Commented May 30, 2021 at 21:07
  • $\begingroup$ Thank you so much! $\endgroup$
    – Stella Dubois
    Commented May 31, 2021 at 16:00

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