Intersection homology cone formula

For a compact $(N-1)$-dimensional link $L$ and traditional perversity $\bar{p}$, finite-chain intersection homology of its open cone is $I{H}_i^{\bar{p}}(cL)=I{H}_i^{\bar{p}}(L)$ for $i<N-1-\bar{p}(N)$ and zero for $i\ge N-1-\bar{p}(N)$. Below the cutoff both chains and possible filling chains avoid the vertex. At and above the cutoff the cone on any allowable cycle is an allowable filling: its new vertex is allowed precisely when $0\le i+1-N+\bar{p}(N)$.

For $\dim L=2n-1$ and lower middle perversity, the suspension with two distinct vertices has $I{H}_i(\mathrm{Susp}L)=I{H}_i(L)$ for $i<n$, zero for $i=n$, and $I{H}_{i-1}(L)$ for $i>n$. Cover it by its two open cones. The Mayer–Vietoris sequence compares their truncated groups with $I{H}_{\ast }(L)$; below $n$ the overlap map is the injective signed diagonal, and above $n$ both cone groups vanish.