gt.geometric topology - Two solid N_3 glued by its boundary

It is a general fact that a closed manifold of odd Euler characteristic cannot bound a compact manifold. This can be deduced pretty easily from the fact that a closed manifold of odd dimension has Euler characteristic zero (a consequence of Poincaré duality) as follows. Suppose N is the boundary of a compact manifold P. Let M be the double of P, the union of two copies of P glued along N. Then the Euler characteristics of M, N, and P are related by:

$chi(M)=chi(P)+chi(P)-chi(N)$

Thus $chi(M)$ and $chi(N)$ are congruent mod 2. If the dimension of N is even, then M is a closed manifold of odd dimension so $chi(M)=0$, hence $chi(N)$ is even. And if the dimension of N is odd then $chi(N)=0$ anyhow.

I should have put this in my book!