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9. Lecture 9: Intersection Homology
9.1. Digression on transversality. Let K ⊂ R be the Cantor set. Let f : R → R be a smooth
(C ∞ ) function that vanishes precisely on K. Let A ⊂ R2 denote the graph of f and let B denote
the x-axis. Then A, B are smooth submanifolds of R2 but their intersection is the Cantor set. This
sort of unruly behavior can be avoided using transversality.
Two submanifolds A, B ⊂ M of a smooth manifold are said to be transverse at a point of their
intersection x ∈ A ∩ B if Tx A + Tx B = Tx M . If A and B are transverse at every point of their
intersection then A ∩ B is a smooth submanifold of M of dimension dim(A) + dim(B) − dim(M ).
Arbitrary submanifolds A, B, ⊂ M can be made to be transverse by moving either one of them,
say A′ = ϕϵ (A) by the flow, for an arbitrarily small time, of a smooth vector field on M . If V is a
finite dimensional vector space of vector fields on M which span the tangent space Tx M at every
point x ∈ M then there is an open and dense subset of V consisting of vector fields v such that
the time = 1 flow ϕ1 of v takes A to a submanifold A′ = ϕ1 (A) that is transverse to B. [This is a
very powerful result. It says, for example, that two submanifolds of Euclidean space can be made
transverse by an arbitrary small translation. The proof, due to Marston Morse, is so elegant, that
I decided to include it here http://www.math.ias.edu/~goresky/math2710/Trans.pdf.]
Two Whitney stratified subsets W1 , W2 ⊂ M are said to be transverse if each stratum of W1 is
transverse to each stratum of W2 , in which case the intersection W1 ∩W2 is also Whitney stratified.
Whitney stratified sets can also be made to be transverse by the application of the flow, for an
arbitrarily small time, of a smooth vector field on M .
9.2. Intersection homology. Let W be a compact n-dimensional Whitney stratified pseudomanifold with strata Sα (α in some index set I, partially ordered by the closure relations between
strata with S0 being the stratum of dimensioin n) and let 0 ≤ pα ≤ cod(Sα ) − 2 be a collection of
integers which we refer to as [trigger warning] a perversity. Define the intersection chains,
9.3. Definition.
(9.3.1)
ICip̄ (W )
{
}
dim(ξ ∩ Sα ) ≤ i − cod(Sα ) + pα
= ξ ∈ Ci (W ) for α > 0
dim(∂ξ ∩ Sα ) ≤ i − 1 − cod(Sα ) + pα
Having placed the same restrictions on the chains as on their boundaries, we obtain a chain
complex, in fact a complex of (soft) sheaves IC p with resulting cohomology groups Hip (W ). (As
usual, “chains” could refer to PL chains, singular chains, subanalytic chains, etc.) Because W is a
pseudomanifold the singular strata have codimension at least 2. The condition pα ≤ cod(Sα ) − 2
implies that most of the chain, and most of its boundary are completely contained within the
top stratum S0 . So a cycle (∂ξ = 0) in ICip̄ is also a cycle for ordinary homology and we have
a homomorphism IHip̄ (W ) → Hi (W ). Moreover, if ξ ∈ ICip̄ (W ) and if η ∈ ICjq̄ (W ) and if we
can arrange that ξ ∩ Sα and η ∩ Sα are transverse within each stratum Sα then we will have an
intersection
p̄+q̄
ξ ∩ η ∈ ICi+j−n
(W )
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which is well defined provided that pα + qα ≤ cod(Sα ) − 2 for all α > 0.
The first problem with this construction is that it is obviously dependent on the stratification.
Moreover, if we are not careful, large values of pα for small strata Sα < Sβ will have the effect
of allowing chains into Sα but not into Sβ thereby “locking” the chain into passing through a
small stratum. This issue can be avoided by requiring that pα depend only on cod(Sα ) and that
β > α =⇒ pβ ≥ pα .
The second problem involves the effect of refining the stratification. For a simple case, suppose
W consists only of two strata, S0 and Sc , the singular stratum having codimension c ≥ 3, to which
we assign a perversity pc . Now suppose we refine this stratum by introducing a “fake” stratum,
Sr of codimension r > c. Chains in ICip (W ) may intersect Sc in dimension ≤ i − c + pc and
for all we know, they may lie completely in Sr , meaning that the chain will have “perversity”
pr = pc + c − r. On the other hand if we assume, as before, that we can arrange for this chain
to be transverse to the fake stratum Sr within the stratum Sc then its intersection with Sr will
have dimension ≤ i − c + pc − (r − c) = i − r + pc which is to say that it has “perversity” pr = pc .
This argument shows (or suggests) that in this case we have natural isomorphisms between the
intersection homology IHipc (W ) as computed before the refinement, and the intersection homology
IHipc ,pr after refinement, for any pr with pc ≤ pr ≤ pc + r − c, that is,
p ,p +1
p ,p +r−c
IHipc ,pc (W ) ∼
(W )
= IHi c c (W ) ∼
= ··· ∼
= IHi c c
In summary, assuming that pc ≤ pr ≤ pc + r − c the resulting homology group IHipc ,pr is unchanged
after refinement. This leads us to the formal definition of intersection homology.
9.4. Definition. A perversity is a function p̄ = (p2 , p3 , · · · ) with p2 = 0 and with pc ≤ pc+1 ≤ pc +1.
The complex of sheaves of intersection chains is the complex with sections
{
}
dim(ξ ∩ Sc ) ≤ i − c + pc
−i
(9.4.1)
Γ(U, IC p̄ ) = ξ ∈ Ci (U ) for c ≥ 2
dim(∂ξ ∩ Sc ) ≤ i − 1 − c + pc
where Sc denotes the union of all strata of codimension c ≥ 2.
Intersection homology with coefficients in a local system is defined similarly, however something
special happens in this case. For any triangulation of a chain ξ ∈ ICp̄−i all of its i-dimensional
simplices and all of its i − 1 dimensional simplices will be completely contained within the top
stratum (or “nonsingular part”) of W . So if L is a local coefficient system defined only on the top
stratum of W , we can still construct the sheaf of intersection chains IC •p̄ (L) exactly as above.
Let 0̄ be the perversity 0c = 0 and let t̄ be the perversity tc = c − 2.
9.5. Theorem. Let W be an oriented stratified pseudomanifold. For any choice of perversity p̄
equation (9.4.1) defines a complex of soft sheaves IC •p̄ on W and the following holds.
(1) The cohomology sheaves IH −m
and the hypercohomology groups IHip̄ (W ) are well defined
p̄
and are independent of the stratification;
(2) in fact they are topological invariants.
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(3) There are canonical maps
H n−i (W ) → IHip̄ (W ) → Hi (W )
that factor the Poincaré duality map,
(4) if p̄ ≤ q̄ then there are also compatible mappings IH p̄ → IH q̄ . In sheaf language we have
natural maps
ZW [n] → IC p̄W → IC q̄W → C •W .
(5) If the link Lx of each stratum is connectred then for p̄ = 0̄ the first of these maps is a
quasi-isomorphism, and for q̄ = t̄ the second map is a quasi-isomorphism.
(6) If pc + qc ≤ tc = c − 2 for all c then the intersection of transversal chains determines a
pairing
p̄+q̄
IHip̄ (W ) × IHjq̄ (W ) → IHi+j−n
(W )
(7) If p̄ + q̄ = t̄ then the resulting pairing
q̄
IHip̄ (W ) × IHn−i
(W ) → H0 (W ) → Z
is nondegenerate over Q (or over any field).
The last statement in Theorem 9.5, Poincaré duality, was the big surprise when intersection
homology was discovered for it is a duality statement that applies to singular spaces. Especially,
if the stratification of W consists only of even codimension strata then there is a “middle” choice
for p, that is, pc = (c − 2)/2 for which IH p̄ (W ; k) is self-dual for any field k.
There is a technical problem with moving chains within a Whitney stratified set W , so as to
be transverse within each stratum of W . This can be accomplished with piecewise-linear chains
within a piecewise-linear stratified set W , and has recently been accomplished using semi-analytic
chains within a semi-analytic stratified set, but to my knowedge, it has not been accomplished
in any other setting. This is one of the many problems that is avoided with the use of sheaf
theory. The proof of topological invariance depends entirely on sheaf theory. Other results such
as the proof of Poincaré duality, that can be established using chain manipulations, are incredibly
awkward, requiring a choice of model for the chains, and delicate manipulations with individual
chains. These constructions are easier, but less geometric, if they are all made using sheaf theory.
For this purpose we need to identify the quasi-isomorphism class of the complex of sheaves IC p̄ .
9.6. Proposition. Let W be a Whitney stratified pseudomanifold and let L be a local coefficient
system defined on the top stratum. Fix a perversity p̄, and let x ∈ Sc be a point in a stratum of
codimension c. Then the stalk of the intersection homology sheaf at x is
{
0
if i < n − pc
H −i (IC p̄ (L))x = IHip̄ (W, W − x; L) =
IHi−n+c−1 (Lx ; L) if i ≥ n − pc
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and the stalk cohomology with compact supports is
Hc−i (Ux ; IC (L)) =
p̄
IHip̄ (Ux )
{
Hi (Lx ; L)
=
0
if i ≤ c − pc
if i > c − pc
Proof. Use the local product structure of a neighborhood Ux ∼
= co (Lx ) × Rn−c and the Künneth
formula
p̄
IHip̄ (U, ∂U ; L) ∼
= IHi−(n−c) (c(Lx ), Lx ; L).
If ξ ∈ ICip̄ and if (i − n + c) − c + pc ≥ 0 then the chain ξ is allowed to hit the cone point, otherwise
it is not. When it is allowed to hit the cone point, we may assume (using a homotopy argument)
that it locally coincides with the cone over a chain in Lx which satisfies the same allowability
conditions. Similar remarks apply to ∂ξ. On the other hand, a compact i-dimensional chain ξ in
the link Lx can be coned down to the cone point it vanishes in cohomology, and such a cone c(ξ)
is allowed provided 0 ≤ (i + 1) − c + pc , that is, if i > c − pc .
□
Comparing this to the calculation (8.3.1) of j∗ j ∗ (IC p̄ ) where j : U = W − Sc → W is the
inclusion of the open complement of the closure of Sc we see that the intersection homology sheaf
on Sc is the truncation of the sheaf j∗ (IC p̄ |U ). For example, suppose dim(W ) = 8 has strata
of dimension 0, 2, 4, 6, 8 and the perversity is the middle one, p(c) = (c − 2)/2. Then the stalk
cohomology H i (C)x of the sheaf IC p̄ looks as follows, where Lr means the r-dimensional link of
the codimension r + 1 stratum and the red zeroes represent homology groups that have been killed
by the perversity condition:
i cod0 cod2
cod4
cod6
0
-1
-2
-3
0
-4
0
-5
0
0
-6
0
IH3 (L5 )
-7
0
IH2 (L3 ) IH4 (L5 )
-8 Z IH1 (L1 ) IH3 (L3 ) IH5 (L5 )
cod8
0
0
0
0
IH4 (L7 )
IH5 (L7 )
IH6 (L7 )
IH7 (L7 )
cod0 cod2
cod4
cod6
cod8
1
3
5
Z IH0 (L ) IH0 (L ) IH0 (L ) IH0 (L7 )
0
IH1 (L3 ) IH1 (L5 ) IH1 (L7 )
0
IH2 (L5 ) IH2 (L7 )
0
0
IH3 (L7 )
0
0
0
0
0
0
Figure 4. Stalk cohomology and compact support cohomology of IC •
This gives an inductive way to construct intersection homology using purely sheaf-theoretic operations, to be described in the next lecture.