= Capped-boundary intersection cohomology
Let a compact oriented $(2m+1)$-manifold $M$ have collared boundary $B$, and cap $B$ by its closed cone. In the lower-middle Deligne convention the result has $IH^i=H^i(M)$ for $i<m$, $IH^m=\operatorname{im}(H^m(M,B)\to H^m(M))$, and $IH^i=H^i(M,B)$ for $i>m$. <Excision> identifies the relative intersection groups with $H^*(M,B)$; the cone neighbourhood has $H^i(B)$ only for $i<m$, so its long exact sequence gives the formula. The cap is Witt exactly when $H^m(B;\mathbb Q)=0$.
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