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It is well-known that odd-dimensional manifolds have Euler characteristiczero. Furthemore orientable manifolds have an even Euler characteristic unlessthe dimension is a multiple of $4$. We prove here a generalisation of thesestatements: a $k$-orientable manifold (or more generally Poincar\'e 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 classvanishes 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 amultiple of $2^k$. For $k=0,1,2,3$, $k$-orientable manifolds with odd Eulercharacteristic exist in all dimensions $2^{k+1}m$, but whether there exist a4-orientable manifold with an odd Euler characteristic is an open question.

Manifolds with odd Euler characteristic and higher orientability