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If a closed oriented $4n$-dimensional manifold $M$ has an orientation-reversing homeomorphism, then the signature $\sigma(M)$ of the intersection form vanishes. More generally, $\sigma(M) = 0$ if $M$ ...

Setup: Consider the braid group $B_n$. One way to define this is as the fundamental group of the unordered configuration space $UC_n(\mathbb{C}) = \{\{z_1,\dotsc,z_n\}\subset \mathbb{C} \mid z_i \not= ...