L² cohomology
Encyclopedia
In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology
except that one uses square-integrable differential form
s. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form
.
L2 cohomology, which grew in part out of L2 d-bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker
(1978) and Jeff Cheeger
(1979). It is closely related to intersection cohomology
; indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology.
Another such result is the Zucker conjecture, which states that for a Hermitian locally symmetric variety the L2 cohomology is isomorphic to the intersection cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker
1982). This was proved in different ways by Looijenga (1988) and by Saper and Stern (1990).
De Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
except that one uses square-integrable differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form
Volume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...
.
L2 cohomology, which grew in part out of L2 d-bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker
Steven Zucker
Steven Mark Zucker is an American mathematician who is well known in the field of mathematics for his work leading to the Zucker conjecture. He is currently part of the mathematics faculty at the Johns Hopkins University.-External links:*...
(1978) and Jeff Cheeger
Jeff Cheeger
Jeff Cheeger , is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and its applications to topology and analysis.-Biography:He graduated from Harvard University with a B.A....
(1979). It is closely related to intersection cohomology
Intersection cohomology
In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.Intersection cohomology was...
; indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology.
Another such result is the Zucker conjecture, which states that for a Hermitian locally symmetric variety the L2 cohomology is isomorphic to the intersection cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker
Zucker
Zucker is a German word meaning "sugar" and may refer to:* Zucker , an album by the Fastbacks* Zucker laboratory rat breedZucker is the surname of:*David Zucker, an American film director...
1982). This was proved in different ways by Looijenga (1988) and by Saper and Stern (1990).