Recent zbMATH articles in MSC 57R20 https://www.zbmath.org/atom/cc/57R20 2021-04-16T16:22:00+00:00 Werkzeug On trivialities of Euler classes of oriented vector bundles over manifolds. https://www.zbmath.org/1456.57025 2021-04-16T16:22:00+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-singh Let $$\mathcal{E}$$ 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)