I am interested in the cohomology ring of the connected sum $M \# N$ of two oriented manifolds $M$ and $N$ in terms of the corresponding cohomology rings of $M$ and $N$.
Mayer-Vietoris shows that in dimensions $0<k<n$ the additive structure is given by
$H^k(M \# N)=H^k(M)\oplus H^k(N)$
via the induced maps of the inclusions. And because this isomorphisms are given by induced maps the cup product translates into componentwise product whenever this is possible, i.e.:
for $([\omega_1],[\eta_1])\in H^i(M)\oplus H^i(N),([\omega_2],[\eta_2])\in H^j(M)\oplus H^j(N)$ AND $i+j<n$.
But how is cup product given if $i+j=n$?
Thanks a lot for your answers!