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).
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. Ast\'erisque, No. 32–33.
Gordon, B. Brent (2001) [1994], "Baily–Borel compactification", Encyclopedia of Mathematics, EMS Presss
Cheeger, Jeff (1983), "Spectral geometry of singular Riemannian spaces", Journal of Differential Geometry, 18 (4): 575–657, MR 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. MR0573430
Cheeger, Jeff On the spectral geometry of spaces with cone-like singularities. Proc. Natl. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106. MR0530173
J. Cheeger, M. Goresky, R. MacPherson, L2 cohomology and intersection homology for singular algebraic varieties, Seminar on differential geometry, vol. 102 of Annals of Mathematics Studies, pages 303–340.MR0645745
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 L2-cohomology of locally symmetric varieties. Compositio Mathematica 67 (1988), no. 1, 3–20. MR0949269
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]. 44. Berlin: Springer-Verlag. ISBN 3-540-43566-2.
Saper, Leslie; Stern, Mark (1990). "L2-cohomology of arithmetic varieties". Annals of Mathematics. (2). 132 (1): 1–69. MR 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 Mathematics 109 (1979), 415–476.
Zucker, Steven, L2-cohomology of warped products and arithmetic groups. Inventiones Mathematicae 70 (1982), 169–218.
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