Intersection homology
For non-singular complex projective algebraic varieties there are a number of (co)homological properties, such as Poincaré duality, Hodge decomposition, hard Lefschetz theorem that are no longer true for the ordinary (co)homology of singular varieties. Intersection (co)homology is a modification of the usual theory designed to retain such properties for the case of singular varieties with, initially, special stress on Poincaré duality in its homological (intersection) form: Let
be a compact oriented
-dimensional manifold, and let
,
be homology classes with representative cycles
and
that intersect in finitely many points (such representation cycles exist). Then the number of points of intersection counted with their multiplicities is independent of
and
, and this defines a duality pairing
.
Let be an
-dimensional complex-analytic variety (possibly with singularities, cf. also Analytic manifold) with a Whitney stratification
. Let
be the codimension of
in
. Let
be one of the usual complexes of geometric chains on
(for instance,
could be the piecewise-linear
-chains with respect to some piecewise-linear structure on
). The complex of intersection chains is defined as the subcomplex
satisfying the condition: a chain
meets each singular stratum
in a set of real dimension
and its boundary intersects each singular stratum
in a set of real dimension
.
The -th intersection homology group
is the
-th homology group of the chain complex
, [a1].
There is also a sheaf-theoretic approach to intersection (co)homology. This involves perverse sheafs [a4], [a5], [a9] (cf. also -module).
There are many applications of intersection (co)homology, in particular to representation theory [a1], [a5], [a8] (for instance, a proof of the Kazhdan–Lusztig conjecture, [a7]).
Another beautiful and very useful property of smooth closed oriented Riemannian (or triangulated) manifolds is that the -th real cohomology group
is isomorphic to the zero eigen space of the appropriate Laplace operator (harmonic cochains). To have something similar for open manifolds, an appropriate "functional cohomology" theory has to be developed. This led to
-cohomology.
Let be any (in general, incomplete) Riemannian manifold with metric
and without boundary. Let
be the space of
-forms on
and let
be exterior differentiation. Let
be the space of square-integrable
-forms with measurable coefficients. The
-cochain complex is now defined by
, and (one definition of) the
-th
-cohomology group of
,
, is as follows:
is the
-th cohomology group of this cochain complex [a2]. In general these cohomology groups depend on the metric
.
Let be again a complex-analytic variety and let
be its non-singular part. In many cases the
-cohomology of
(with respect to an appropriate metric) has been found to be the dual of the intersection homology of
, [a1]–[a3], [a10], [a11].
References
[a1] | R.D. MacPherson, "Global questions in the topology of singular spaces" , Proc. Internat. Congress Mathematicians (Warszawa, 1983) , 1 , PWN & Elsevier (1984) pp. 213–236 |
[a2] | J. Cheeger, M. Goresky, R.D. MacPherson, "![]() |
[a3] | S. Zucker, "![]() |
[a4] | A. Borel, et al., "Intersection cohomology" , Birkhäuser (1984) MR0788176 Zbl 0553.14002 |
[a5] | "Analyse et topologie sur les espaces singuliers I-III" Astérisque , 100–102 (1982) |
[a6] | F.C. Kirwan, "An introduction to intersection homology theory" , Longman (1988) MR0981185 Zbl 0656.55002 |
[a7] | J.-L. Brylinski, M. Kashiwara, "Kazhdan–Lusztig conjecture and holonomic systems" Invent. Math. , 64 (1981) pp. 387–410 MR0632980 Zbl 0473.22009 |
[a8] | T.A. Springer, "Perverse sheafs and representation theory" P. Fong (ed.) , The Arcata Conf. Representations of Finite Groups , Proc. Symp. Pure Math. , 1 , Amer. Math. Soc. (1987) pp. 315–322 MR0933368 |
[a9] | J.-L. Brylinski, "(Co-)homologie d'intersection et faisceaux pervers (Sem. Bourbaki 1981/1982, Exp. 585)" Astérisque , 92–93 (1982) pp. 129–158 |
[a10] | E. Looyenga, "![]() |
[a11] | L. Saper, M. Stern, "![]() |
Intersection homology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intersection_homology&oldid=24477