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 2k+1⁠, where we call a manifold k-orientable if the i-th Stiefel–Whitney class vanishes for all 0<i