Intersection homology ($I{H}_{\ast }^{\bar{p}}(X)$)

Intersection homology is the homology of a chain complex in which chains and their boundaries satisfy codimension-dependent perversity restrictions at singular strata. Unless specified otherwise coefficients here are $\mathbb{Q}$. Ordinary homology is recovered on manifolds.

A traditional perversity is a function $\bar{p}:\{2,3,\dots \}\to \mathbb{Z}$ satisfying $\bar{p}(2)=0$ and $\bar{p}(k+1)-\bar{p}(k)\in \{0,1\}$. It specifies how far chains may meet strata of codimension $k$.

An $i$-dimensional singular simplex $\sigma :{\Delta }^i\to X$ is $\bar{p}$-allowable if ${\sigma }^{-1}(X_{N-k})$ is contained in the $(i-k+\bar{p}(k))$-skeleton of ${\Delta }^i$ for every $k\ge 2$. A negative skeleton is empty. Individual faces need not be allowable; this is why the definition of intersection chains separately imposes allowability on the total boundary.

The group $I{S}_i^{\bar{p}}(X)$ consists of finite singular chains whose nonzero simplices are allowable singular simplices and whose boundary, after cancellation, also consists of allowable simplices. Since ${\partial }^2=0$, these groups form a chain complex. Its homology is intersection homology.

The upper middle perversity is $\bar{n}(k)=\lceil (k-2)/2\rceil$. It differs from the lower middle perversity by one at odd codimensions, and is complementary to it.

The lower middle perversity is $\bar{m}(k)=\lfloor (k-2)/2\rfloor$. The complementary upper middle perversity is $\bar{n}(k)=\lceil (k-2)/2\rceil$, and $\bar{m}(k)+\bar{n}(k)=k-2$. They agree at even codimensions.

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.

In the Deligne-sheaf convention, $I{H}_{\bar{p}}^i(X)={\mathbb{H}}^i(X,I{C}_{\bar{p}})$, with $I{C}_{\bar{p}}$ normalized in degree zero on the regular stratum. For a compact oriented $N$-dimensional pseudomanifold over a field, $I{H}_{\bar{p}}^i(X)\cong I{H}_{N-i}^{\bar{p}}(X)$ and it is dual to $I{H}_i^{\bar{q}}(X)$, where $\bar{p}(k)+\bar{q}(k)=k-2$. Thus the dual of lower-middle singular chains uses the upper-middle Deligne sheaf. The two middle conventions agree on Witt spaces.

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 $I{H}^i=H^i(M)$ for $i<m$, $I{H}^m=\mathrm{im}(H^m(M,B)\to H^m(M))$, and $I{H}^i=H^i(M,B)$ for $i>m$. Excision identifies the relative intersection groups with $H^{\ast }(M,B)$; 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 $H^m(B;\mathbb{Q})=0$.

The degree-zero normalized Deligne intersection complex extends the constant sheaf from the regular stratum by successive derived direct images and cohomological truncations at degree $\bar{p}(k)$ when codimension-$k$ strata are added. Its hypercohomology defines intersection cohomology. On a cone with vertex of codimension $k$, its local cohomology is the link intersection cohomology in degrees at most $\bar{p}(k)$ and zero above.

For an isolated singular point of an $N$-dimensional oriented pseudomanifold, the unshifted middle intersection complex restricts to $\mathbb{Q}$ on the regular stratum and has no negative cohomology. Its support condition is $H^j(i_x^{\ast }I{C}_{\bar{p}})=0$ for $j>\bar{p}(N)$; its cosupport condition is $H^j(i_x^{!}I{C}_{\bar{p}})=0$ for $j\le \bar{p}(N)+1$. These conditions, together with constructibility and normalization, characterize it. On a Witt space they are exchanged by $IC\cong D(IC)[-N]$.