Odd-dimensional orbifolds with all geodesics closed are covered by manifolds

Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.

A note on the Kervaire semi-characteristic

We present a potential generalization of the Kervarie semi-characteristic (with real coefficient) to the case of non-orientable manifolds.

Euler classes of vector bundles over manifolds

Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

Manifolds with Odd Euler Characteristic and Higher Orientability

It is well known that odd-dimensional manifolds have Euler characteristic zero. Furthermore, orientable manifolds have an even Euler characteristic unless the dimension is a multiple of $4$. We prove here a generalisation of these statements: a $k$-orientable manifold (or more generally Poincaré complex) has even Euler characteristic unless the dimension is a multiple of $2^{k+1}$, where we call a manifold $k$-orientable if the i-th Stiefel–Whitney class vanishes for all $0<i< 2^k$ ($k\geq 0$). More generally, we show that for a $k$-orientable manifold the Wu classes $v_l$ vanish for all $l$ that are not a multiple of $2^k$. For $k=0,1,2,3$, $k$-orientable manifolds with odd Euler characteristic exist in all dimensions $2^{k+1}m$, but whether there exists a 4-orientable manifold with an odd Euler characteristic is an open question.

Linking numbers in non-orientable 3-manifolds

The construction of integer linking numbers of closed curves in a three-dimensional manifold usually appeals to the orientation of this manifold. We discuss how to avoid it constructing similar homotopy invariants of links in non-orientable manifolds.