Timeline for Possible Euler characteristics of manifolds with tangential structures

7 events

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Jan 22 at 9:29	comment	added	Simona Vesela		@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.
Jan 19 at 10:49	comment	added	John Rognes		Have you seen the related work of Kreck and Zagier? mathoverflow.net/questions/388956/…
Jan 19 at 10:34	history	edited	Simona Vesela	CC BY-SA 4.0	incorporated comments
Jan 19 at 10:29	comment	added	Simona Vesela		@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 $p^*w_{2k}$.
Jan 18 at 14:53	comment	added	Michael Albanese		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 $w_{2n}$.
S Jan 18 at 12:20	review	First questions
Jan 18 at 15:09
S Jan 18 at 12:20	history	asked	Simona Vesela	CC BY-SA 4.0