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 :).



Cobordisms and Euler characteristics
René Thom proved that two closed $n$-manifolds $M$ and $N$ are (unoriented) cobordant iff their [Stiefel-Whitney numbers](https://en.wikipedia.org/wiki/Stiefel%E2%80%93Whitney_class#Stiefel.E2.80.93Whitney_numbers) agree: for any partition $i\_1 + \dotsb + i\_k = n$,


$$[M]\frown w\_{i-1}(M)w\_{i\_2}(M)\dotsm w\_{i\_k}(M) = [N]\frown w\_{i-1}(N)w\_{i\_2}(N)\dotsm w\_{i\_k}(N).$$


The mod 2 Euler characteristic is a Stiefel-Whitney number: $\chi(M)\bmod 2 = [M]\frown w\_n(M)$. Thus, if $M$ and $N$ are unoriented cobordant, then their Euler characteristics differ by a multiple of 2. (There might be a more direct proof of this using handles.)


You can get rid of the mod 2 assumption if you work with (stably almost) complex cobordism, equipping all manifolds and cobordisms with a complex structure on the stable normal bundle. (In particular, this determines an orientation of your manifold.) Milnor and Novikov showed that $M$ and $N$ are complex cobordant iff their Chern numbers agree: these are defined in the same way as Stiefel-Whitney numbers, but for Chern classes.


The Euler characteristic is a Chern number: $\chi(M) = [M]\frown c\_n(M)$ if $M$ is $2n$-dimensional, and $\chi(M) = [M]\frown 0$ if $M$ is odd-dimensional. Thus in any dimension, the Euler characteristic is a cobordism invariant for stably almost complex cobordism.


