I am trying to understand exactly which role the Euler characteristic plays in (smooth) cobordism theory, and especially why the answer seems to depend on the dimensions of the manifolds in question. Suppose $M$ and $N$ are two closed smooth $n$-manifolds and that we have a smooth cobordism $(W;M,N)$ (i.e. $W$ is a smooth compact $(n+1)$-manifold with boundary $\partial W=M\sqcup N$). Poincare-Lefschetz duality then predicts that $H_k(W,N;\mathbb{Z}_2)\cong H_{n+1-k}(W,M;\mathbb{Z}_2)$ for every $k\in \mathbb{Z}$. In particular, if we denote by $\chi(W,M)$ the Euler characteristic of the homology $H_{*}(W,M;\mathbb{Z}_2)$ we see that $\chi(W,N)=(-1)^{n+1}\chi(W,M)$. By additivity of the Euler characteristic we also have $$ \chi(W,N)+\chi(N)=\chi(W)=\chi(W,M)+\chi(M). $$ If $n$ is odd we therefore conclude that $\chi(M)=\chi(N)$, so in this case $\chi$ is a cobordism invariant. My question is:

What happens when $n$ is even? Is the Euler useful in any way for understanding cobordisms of even dimensional manifolds? Any interesting statement would be much appreciated, even if additional assumptions such as orientability, spin etc. are needed. (Note that the sphere $S^2$ is null-cobordant, the cobordism given by the 3-ball, so clearly $\chi$ is not a cobordism invariant of even dimensional manifolds...)

Thanks in advance :).