Recent zbMATH articles in MSC 57R20https://www.zbmath.org/atom/cc/57R202021-04-16T16:22:00+00:00WerkzeugOn trivialities of Euler classes of oriented vector bundles over manifolds.https://www.zbmath.org/1456.570252021-04-16T16:22:00+00:00"Naolekar, Aniruddha C."https://www.zbmath.org/authors/?q=ai:naolekar.aniruddha-c"Subhash, B."https://www.zbmath.org/authors/?q=ai:subhash.bhaskaran"Thakur, Ajay Singh"https://www.zbmath.org/authors/?q=ai:thakur.ajay-singhLet \(\mathcal{E}\) be the set of diffeomorphism classes of closed connected smooth manifolds \(X \) such that for every oriented vector bundle \(\alpha\) over \(X\), the Euler class \(e(\alpha) = 0\), and let \(\mathcal{E}_k\) be the subset of \(\mathcal{E}\) of manifolds with dimension \(k\). In the recent literature there are results about the same subject by considering oriented bundles over \(CW\)-complexes.
In this very interesting paper, the authors give a complete description for \(k \leq 5\) and give partial results for \(k= 6\). Let us denote by \(X\) a closed connected smooth \(k\)-manifold. Then one has the following results:
\begin{itemize}
\item \(X \in \mathcal{E}_3\) if and only if \(X\) is a smooth homology \(3\)-sphere.
\item \( \mathcal{E}_4 = \emptyset\).
\item \(X \in \mathcal{E}_5\) if and only if \(H^2(X;\mathbb{Z}) = 0; \, H^4(X;\mathbb{Z} ) = 0\), and the \(2\)-primary component of \(H_2(X; \mathbb{Z})\) is trivial
\end{itemize}
There exists an oriented vector bundle \(\alpha\) over the projective space \(\mathbb{RP}^5\) or over the lens spaces of dimension \( 5\) such that \( e(\alpha) \neq 0\). Then the authors construct examples of \(X \in \mathcal{E}_5 \) in the case where \(X\) is a simply connected closed smooth \(5\)-manifold, for \(X\) a smooth closed non-orientable manifold and also for \(X\) a non-simply connected closed smooth oriented \(5\)-manifold.
The authors also observe that any \(X \in \mathcal{E}_6 \) must have non-positive Euler characteristic.
Reviewer: Alice Kimie Miwa Libardi (São Paulo)