Hej,

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.:

$$([\omega_1],[\eta_1])\cup([\omega_2],[\eta_2])=([\omega_1\cup\omega_2],[\eta_1\cup\eta_2])$$

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!