Let $M, N$ be two compact oriented manifolds with dimensions $4m$ resp $4n$. Is there something we can say about the signature $\sigma (M \times N)$, maybe even a formula only depending on $ \sigma (M), \sigma (N)$ and $m, n$?
I tried to apply the Künneth formula, but this only resulted in a lenghty and confusing computation.
In fact, $\sigma(M\times N) = \sigma(M)\sigma(N)$. (The result also holds without the restriction than $\dim M, \dim N$ are divisible by $4$, though it's almost vacuous in that case.) To prove it, use the Künneth formula to show that the only contributions to the intersection form on $M\times N$ are of the form $H^{2m}(M)\otimes H^{2n}(N)$, then decompose it into eigenvectors of the Poincare duality map on $H^{2m}(M)$ and $H^{2n}(N)$.