mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, L² cohomology is a

cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

for smooth non-compact

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

s ''M'' with

Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

. It is defined in the same way as

de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) 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 adapte ...

except that one uses

square-integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...

differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...

s. The notion of square-integrability makes sense because the

metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

on ''M'' gives rise to a

norm Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...

on differential forms and a

volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...

. 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 Jeff Cheeger (born December 1, 1943) is an American mathematician and Silver Professor at the Courant Institute of Mathematical Sciences of New York University. His main interest is differential geometry and its connections with topology and an ...

(1979). It is closely related to

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 ov ...

; 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 {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...

locally symmetric variety the L² cohomology is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to the

(with the

middle perversity 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 ov ...

) of its Baily–Borel compactification (Zucker 1982). This was proved in different ways by

Eduard Looijenga Eduard Jacob Neven Looijenga (born 30 September 1948, Zaandam) is a Dutch mathematician who works in algebraic geometry and the theory of algebraic groups.

(1988) and by Leslie Saper and Mark Stern (1990).

References

* * * * * * *

Mark Goresky Robert Mark Goresky is a Canadian mathematician who invented intersection homology with his advisor and life partner Robert MacPherson. Career Goresky received his Ph.D. from Brown University in 1976. His thesis, titled ''Geometric Cohomology ...

L² cohomology is intersection cohomology
* Frances Kirwan, Jonathan Woolf ''An Introduction to Intersection Homology Theory,'', chapter 6 * * * * * * Cohomology theories Differential geometry Differential topology {{topology-stub

See also

References