# Manifolds with odd Euler characteristic and higher orientability

## TL;DR

This paper generalizes classical results on Euler characteristics of manifolds, showing that $k$-orientable manifolds have even Euler characteristic unless their dimension is a multiple of $2^{k+1}$, with some cases remaining open.

## Contribution

It introduces the concept of $k$-orientability for manifolds and establishes new conditions relating $k$-orientability to Euler characteristic parity and Wu classes.

## Key findings

- $k$-orientable manifolds have even Euler characteristic unless dimension is multiple of $2^{k+1}$.
- Wu classes vanish for all $l$ not multiple of $2^k$ in $k$-orientable manifolds.
- Existence of odd Euler characteristic in $k$-orientable manifolds for certain $k$ and dimensions is characterized, with some open questions.

## Abstract

It is well-known that odd-dimensional manifolds have Euler characteristic zero. Furthemore 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 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 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 exist a 4-orientable manifold with an odd Euler characteristic is an open question.