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L² cohomology

March 13, 2024

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

See also edit

References edit

  • Atiyah, Michael F. (1976). "Elliptic operators, discrete groups and von Neumann algebras". Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974). Paris: Soc. Math. France. pp. 43–72. Astérisque, No. 32–33.
  • Gordon, B. Brent (2001) [1994], "Baily–Borel compactification", Encyclopedia of Mathematics, EMS Press
  • Cheeger, Jeff (1983), "Spectral geometry of singular Riemannian spaces", Journal of Differential Geometry, 18 (4): 575–657, doi:10.4310/jdg/1214438175, MR 0730920
  • Cheeger, Jeff (1980). "On the Hodge theory of Riemannian pseudomanifolds". Geometry of the Laplace operator. Proc. Sympos. Pure Math. Vol. 36. Providence, R.I.: American Mathematical Society. pp. 91–146. MR 0573430.
  • Cheeger, Jeff (1979). "On the spectral geometry of spaces with cone-like singularities". Proc. Natl. Acad. Sci. U.S.A. 76 (5): 2103–2106. Bibcode:1979PNAS...76.2103C. doi:10.1073/pnas.76.5.2103. MR 0530173. PMC 383544. PMID 16592646.
  • Cheeger, J.; Goresky, M.; MacPherson, R. "L2 cohomology and intersection homology for singular algebraic varieties". Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. pp. 303–340. MR 0645745.
  • Mark Goresky, L2 cohomology is intersection cohomology
  • Frances Kirwan, Jonathan Woolf An Introduction to Intersection Homology Theory,, chapter 6 ISBN 1-58488-184-4
  • Looijenga, Eduard (1988). "L2-cohomology of locally symmetric varieties". Compositio Mathematica. 67 (1): 3–20. MR 0949269.
  • Lück, Wolfgang (2002). L2-invariants: theory and applications to geometry and K-theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Vol. 44. Berlin: Springer-Verlag. ISBN 3-540-43566-2.
  • Saper, Leslie; Stern, Mark (1990). "L2-cohomology of arithmetic varieties". Annals of Mathematics. Second Series. 132 (1): 1–69. doi:10.2307/1971500. JSTOR 1971500. MR 1059935.
  • Zucker, Steven (1978). "Théorie de Hodge à coefficients dégénérescents". Compt. Rend. Acad. Sci. 286: 1137–1140.
  • Zucker, Steven (1979). "Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré metric". Annals of Mathematics. 109 (3): 415–476. doi:10.2307/1971221. JSTOR 1971221.
  • Zucker, Steven (1982). "L2-cohomology of warped products and arithmetic groups". Inventiones Mathematicae. 70 (2): 169–218. Bibcode:1982InMat..70..169Z. doi:10.1007/BF01390727. S2CID 121348276.


 

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March 13, 2024
cohomology, mathematics, cohomology, cohomology, theory, smooth, compact, manifolds, with, riemannian, metric, defined, same, rham, cohomology, except, that, uses, square, integrable, differential, forms, notion, square, integrability, makes, sense, because, m. 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 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 Eduard Looijenga 1988 and by Leslie Saper and Mark Stern 1990 See also editDirichlet form Dirichlet principle Riemannian manifoldReferences editAtiyah Michael F 1976 Elliptic operators discrete groups and von Neumann algebras Colloque Analyse et Topologie en l Honneur de Henri Cartan Orsay 1974 Paris Soc Math France pp 43 72 Asterisque No 32 33 Gordon B Brent 2001 1994 Baily Borel compactification Encyclopedia of Mathematics EMS Press Cheeger Jeff 1983 Spectral geometry of singular Riemannian spaces Journal of Differential Geometry 18 4 575 657 doi 10 4310 jdg 1214438175 MR 0730920 Cheeger Jeff 1980 On the Hodge theory of Riemannian pseudomanifolds Geometry of the Laplace operator Proc Sympos Pure Math Vol 36 Providence R I American Mathematical Society pp 91 146 MR 0573430 Cheeger Jeff 1979 On the spectral geometry of spaces with cone like singularities Proc Natl Acad Sci U S A 76 5 2103 2106 Bibcode 1979PNAS 76 2103C doi 10 1073 pnas 76 5 2103 MR 0530173 PMC 383544 PMID 16592646 Cheeger J Goresky M MacPherson R L2 cohomology and intersection homology for singular algebraic varieties Seminar on Differential Geometry Annals of Mathematics Studies Vol 102 pp 303 340 MR 0645745 Mark Goresky L2 cohomology is intersection cohomology Frances Kirwan Jonathan Woolf An Introduction to Intersection Homology Theory chapter 6 ISBN 1 58488 184 4 Looijenga Eduard 1988 L2 cohomology of locally symmetric varieties Compositio Mathematica 67 1 3 20 MR 0949269 Luck Wolfgang 2002 L2 invariants theory and applications to geometry andK theory Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge A Series of Modern Surveys in Mathematics Results in Mathematics and Related Areas 3rd Series A Series of Modern Surveys in Mathematics Vol 44 Berlin Springer Verlag ISBN 3 540 43566 2 Saper Leslie Stern Mark 1990 L2 cohomology of arithmetic varieties Annals of Mathematics Second Series 132 1 1 69 doi 10 2307 1971500 JSTOR 1971500 MR 1059935 Zucker Steven 1978 Theorie de Hodge a coefficients degenerescents Compt Rend Acad Sci 286 1137 1140 Zucker Steven 1979 Hodge theory with degenerating coefficients L2 cohomology in the Poincare metric Annals of Mathematics 109 3 415 476 doi 10 2307 1971221 JSTOR 1971221 Zucker Steven 1982 L2 cohomology of warped products and arithmetic groups Inventiones Mathematicae 70 2 169 218 Bibcode 1982InMat 70 169Z doi 10 1007 BF01390727 S2CID 121348276 nbsp This differential geometry related article is a stub You can help Wikipedia by expanding it vte nbsp This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title L cohomology amp oldid 1094074841, wikipedia, wiki, book, books, library,

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