= Fixed-point congruence for an involution on an odd-sphere connected sum
{title2=$\#\operatorname{Fix}(f)\equiv2-2g\pmod4$}
Let $W_g$ be a <connected sum of oriented manifolds> consisting of $g$ copies of $S^r\times S^r$ with odd $r\geq1$. Suppose an orientation-preserving smooth <involution> $f$ has finitely many fixed points, all nondegenerate with positive <determinant> $\det(I-Df_x)$. The <Lefschetz-Hopf fixed-point theorem> and the <cohomology ring of a connected sum of odd-dimensional sphere products> give
$$
\#\operatorname{Fix}(f)=L(f)=2-\operatorname{tr}(f^*|_{H^r(W_g;\mathbb R)}).
$$
This middle group has dimension $2g$ and its <Poincare duality pairing> is alternating. The induced map preserves the pairing and squares to the identity. By <eigenspaces of a symplectic involution>, its trace is $2g-4b$ for an integer $b$. Therefore $\#\operatorname{Fix}(f)\equiv2-2g\pmod4$.
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