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Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even dimension dimension, a manifold with odd Euler characteristics. Note that Euler characteristic of $2$ can always be achieved, using the sphere $S^{2k}$.

For example, in the case of $BSpin\to BO$, the odd Euler characteristics can be achieved in dimensions $0\pmod{8}$ (for example the quaternionic projective plane HP^2) and cannot be achieved in all other dimensions $\pmod{8}$.

One possible approach might be through characteristics classes. Recall that for a $2k$-dimensional manifold $M$, the middle Wu class $v_k\in H^k(M;\mathbb{Z}/2)$ satisfies the following equation

$$x^2=v_kx\quad \forall x\in H^k(M;\mathbb{Z}/2)$$

in particular we know that if $v_k=0$, then the intersection form of the manifold $M$ is even, i.e. $\langle x^2,[M]\rangle\equiv 0$, then the middle cohomology has an even rank and so the Poincare duality gives that $M$ necessarily has even Euler characteristisc.

EDIT: as prof. Albanese suggested the same conclustion can be achieved if $p^*w_{2k}=0$ in $H^{2k}(B;\mathbb{Z}/2)$ because $\chi(M)\equiv \langle w_{2k}(M),[M]\rangle\pmod{2}$.

Now we might be able to prove that a specific $B$-structure on a manifold forces the class $v_k$ to vanish, as was the strategy of the paper https://arxiv.org/abs/1704.06607.

My question is: In the cases, where the above method does not help, how could one try to solve the question for a given $B$?

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    $\begingroup$ If $p : B \to BO$ denotes the fibration map, then a necessary condition for there to exist a $2n$-dimensional closed manifold with $B$-structure and odd Euler characteristic is that $p^*w_{2n} \in H^{2n}(B; \mathbb{Z}_2)$ is non-zero. This is because the Euler characteristic on a closed $2n$-dimensional manifold is determined by the Euler class which reduces mod $2$ to $w_{2n}$. $\endgroup$
    – Michael Albanese
    Commented Jan 18 at 14:53
  • $\begingroup$ @MichaelAlbanese Thank you for an answer. Yes, it is connected to the Wu class in a sense that $v_k^2=w_{2k}$ in the manifold. I will edit the question because this seems like a useful point to consider $p^*w_{2k}$. $\endgroup$
    – Simona Vesela
    Commented Jan 19 at 10:29
  • $\begingroup$ Have you seen the related work of Kreck and Zagier? mathoverflow.net/questions/388956/… $\endgroup$
    – John Rognes
    Commented Jan 19 at 10:49
  • $\begingroup$ @JohnRognes Thank you for a response. I have attended a talk about it, and I remember thinking that it would be hard to show that the manifolds they construct have any particular $B$-structure, because Kreck-Zagier work only rationally. But I'll check my notes to see. $\endgroup$
    – Simona Vesela
    Commented Jan 22 at 9:29

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