L² cohomology

L² cohomology

In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds "M" with Riemannian metric. It defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. 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).

References

*springer|id=B/b130010|author=B. Brent Gordon|title=Baily-Borel compactification
*Cheeger, Jeff "Spectral geometry of singular Riemannian spaces." J. Differential Geom. 18 (1983), no. 4, 575--657 (1984).MathSciNet|id=0730920
*Cheeger, Jeff "On the Hodge theory of Riemannian pseudomanifolds." Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 91--146, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980. MathSciNet|id=0573430
*Cheeger, Jeff "On the spectral geometry of spaces with cone-like singularities." Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103--2106. MathSciNet|id=0530173
*J. Cheeger, M. Goresky, R. MacPherson, "L2 cohomology and intersection homology for singular algebraic varieties", Seminar on differential geometry, vol. 102 of Annals of mathamtical studies, pages 303-340.MathSciNet|id=0645745
*M. Goresky [http://www.math.ias.edu/~goresky/pdf/zucker.pdf L2 cohomology is intersection cohomology ]
*Frances Kirwan, Jonathan Woolf "An Introduction to Intersection Homology Theory,", chapter 6 ISBN 1584881844
*Looijenga, Eduard "L2-cohomology of locally symmetric varieties." Compositio Math. 67 (1988), no. 1, 3-20. MathSciNet|id=0949269
*Saper, Leslie; Stern, Mark "L2-cohomology of arithmetic varieties." Ann. of Math. (2) 132 (1990), no. 1, 1-69. MathSciNet|id=1059935
*Zucker, Steven, "Théorie de Hodge à coefficients dégénérescents." Comptes Rendus Acad. Sci. 286 (1978), 1137-1140.
*Zucker, Steven, "Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré metric." Annals of Math. 109 (1979), 415-476.
*Zucker, Steven, "L2-cohomology of warped products and arithmetic groups." Inventiones Math. 70 (1982), 169-218.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Cohomology — In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries.… …   Wikipedia

  • Cohomology operation — In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a… …   Wikipedia

  • Cohomology ring — In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here cohomology is usually understood …   Wikipedia

  • Cohomology with compact support — In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support. de Rham cohomology with compact support for smooth manifolds Given a manifold… …   Wikipedia

  • Cohomology of algebras — In mathematics, the homology or cohomology of an algebra may refer to Banach algebra cohomology of a bimodule over a Banach algebra Cyclic homology of an associative algebra Group cohomology of a module over a group ring or a representation of a… …   Wikipedia

  • cohomology — noun Date: circa 1959 a part of the theory of topology in which groups are used to study the properties of topological spaces and which is related in a complementary way to homology theory called also cohomology theory • cohomological adjective …   New Collegiate Dictionary

  • cohomology theory — noun see cohomology …   New Collegiate Dictionary

  • cohomology — noun a) A theory associating a system of quotient groups to each topological space. b) A system of quotient groups associated to a topological space …   Wiktionary

  • Étale cohomology — In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil… …   Wikipedia

  • Group cohomology — This article is about homology and cohomology of a group. For homology or cohomology groups of a space or other object, see Homology (mathematics). In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well… …   Wikipedia

  • Crystalline cohomology — In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by Alexander Grothendieck (1966, 1968) and developed by Pierre Berthelot (1974). Its values are modules over rings of Witt vectors over the base… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”