Let $M$ be a compact oriented $4k$-manifold, $\chi_M$ the euler characteristic of $M$ and $sig(M)$ the signature of $M$. Why is $\chi_M \equiv sig(M)$ mod $2$?
In the book " a concise course in algebraic topology" from J.P. May on page 166/167 there is a proof which I don't understand and which is very short. I will repeat the proof here:
It is $sig(M)=r-s$, where $r+s=dim H^{2k}(M;\mathbb{R})\equiv \chi_M$ mod $2$.
I only understand that it is $sig(M)=r-s$ with $r+s=dim H^{2k}(M;\mathbb{R})$. But everything else I don't understand. Could you explain me the proof or do you have an other idea?
By Poincare duality the Betti numbers for cohomology groups of complementary dimensions all cancel if one works modulo 2. That leaves one with the middle dimension $2k$. Again working modulo 2, one has $r+s=r-s$, whence the result.