L² cohomology

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

• Dirichlet form
• Dirichlet principle
• Riemannian manifold

References

• 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.