= Cohomology ring of a connected sum of odd-dimensional sphere products
{title2=$H^*(W_g;\mathbb Z)$}
For odd $r\geq1$ and $W_g=\mathbin\#_{i=1}^g(S^r\times S^r)$, the <integral cohomology> has one copy of $\mathbb Z$ in degrees zero and $2r$, $\mathbb Z^{2g}$ in degree $r$, and zero elsewhere. Choose the top <orientation class> $\omega$ and degree-$r$ classes $\alpha_i,\beta_i$ from the two factors of summand $i$. Their nonzero positive-degree basis products are
$$
\alpha_i\smile\beta_j=\delta_{ij}\omega,\qquad \beta_j\smile\alpha_i=-\delta_{ij}\omega.
$$
All $\alpha_i\alpha_j$ and $\beta_i\beta_j$ vanish, and $\omega$ annihilates positive-degree classes. The <Künneth theorem> computes each sphere product; the <Mayer–Vietoris sequence> computes the connected sum groups, and its degree-one <pinch maps> determine the products. Thus its middle-degree <Poincare duality pairing> is a <symplectic vector space> after changing coefficients to a field of characteristic different from two. The case $r=1$ is the usual <cohomology ring of a closed oriented surface>.
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