View source for L² cohomology

In [[mathematics]], '''L<sup>2</sup> cohomology''' is a [[cohomology theory]] for [[smooth manifold|smooth]] non-compact [[manifold]]s ''M'' with [[Riemannian metric]]. It is defined in the same way as [[de Rham cohomology]] except that one uses [[square-integrable]] [[differential form]]s. The notion of square-integrability makes sense because the [[metric (mathematics)|metric]] on ''M'' gives rise to a [[norm (mathematics)|norm]] on differential forms and a [[volume form]].

L<sup>2</sup> cohomology, which grew in part out of L<sup>2</sup> 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 variety|Hermitian]] [[locally symmetric variety]] the L<sup>2</sup> 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==
{{refbegin|2}}
*{{Cite book| publisher = Soc. Math. France
| pages = 43–72. Astérisque, No. 32–33
| last = Atiyah
| first = Michael F.
| author-link=Michael Atiyah
| title = Colloque "Analyse et Topologie" en l'Honneur de [[Henri Cartan]] (Orsay, 1974)
| chapter = Elliptic operators, discrete groups and von Neumann algebras
| location = Paris
| year = 1976
}}
*{{springer|id=B/b130010|first=B. Brent|last= Gordon|title=Baily–Borel compactification}}
*{{citation|last=Cheeger|first= Jeff|author-link=Jeff Cheeger| title=Spectral geometry of singular Riemannian spaces|journal= [[Journal of Differential Geometry]]|volume= 18 |year=1983|issue= 4|pages= 575–657|doi= 10.4310/jdg/1214438175|mr=0730920|doi-access= free}}
*{{cite book |last=Cheeger |first=Jeff |chapter=On the Hodge theory of Riemannian pseudomanifolds |title=Geometry of the Laplace operator |pages=91–146 |series=Proc. Sympos. Pure Math. |volume=36 |publisher=American Mathematical Society |location=Providence, R.I. |year=1980 |mr=0573430}}
*{{cite journal |last=Cheeger |first=Jeff |title=On the spectral geometry of spaces with cone-like singularities |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=76 |year=1979 |issue=5 |pages=2103–2106 |doi=10.1073/pnas.76.5.2103 |pmid=16592646 |pmc=383544 |bibcode=1979PNAS...76.2103C |mr=0530173}}
*{{cite book |last1=Cheeger |first1=J. |last2=Goresky |first2=M. |last3=MacPherson |first3=R. |chapter=L<sup>2</sup> cohomology and intersection homology for singular algebraic varieties |title=Seminar on Differential Geometry |volume=102 |series=Annals of Mathematics Studies |pages=303–340 |mr=0645745}}
*[[Mark Goresky]], [http://www.math.ias.edu/~goresky/pdf/zucker.pdf L<sup>2</sup> cohomology is intersection cohomology ]
*[[Frances Kirwan]], Jonathan Woolf ''An Introduction to Intersection Homology Theory,'', chapter 6 {{isbn|1-58488-184-4}}
*{{cite journal |author-link=Eduard Looijenga |last=Looijenga |first=Eduard |title=L<sup>2</sup>-cohomology of locally symmetric varieties |journal=[[Compositio Mathematica]] |volume=67 |year=1988 |issue=1 |pages=3–20 |mr=0949269}}
*{{Cite book
| publisher = Springer-Verlag
| isbn = 3-540-43566-2
| volume = 44
| last = Lück
| first = Wolfgang
| author-link=Wolfgang Lück
| title = L<sup>2</sup>-invariants: theory and applications to geometry and ''K''-theory
| location = Berlin
| series = 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]
| year = 2002
}}
*{{cite journal|last1=Saper|first1= Leslie|author2-link=Mark Stern|last2=Stern|first2= Mark|title= L<sub>2</sub>-cohomology of arithmetic varieties|journal= [[Annals of Mathematics]]|series=Second Series |volume= 132 |year=1990|issue=1|pages= 1–69|doi= 10.2307/1971500|jstor= 1971500|mr=1059935}}
*{{cite journal|last1=Zucker |first1= Steven |title=Théorie de Hodge à coefficients dégénérescents |journal=Compt. Rend. Acad. Sci. |volume=286 |year=1978 |pages=1137–1140}}
*{{cite journal|last1=Zucker |first1= Steven |title=Hodge theory with degenerating coefficients: L<sup>2</sup>-cohomology in the Poincaré metric |journal=[[Annals of Mathematics]] |volume=109 |year=1979 |issue= 3 |pages=415–476|doi= 10.2307/1971221 |jstor= 1971221 }}
*{{cite journal|last1=Zucker |first1= Steven |title=L<sup>2</sup>-cohomology of warped products and arithmetic groups |journal=[[Inventiones Mathematicae]] |volume=70 |year=1982 |issue= 2 |pages=169–218|doi= 10.1007/BF01390727 |bibcode= 1982InMat..70..169Z |s2cid= 121348276 }}
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{{DEFAULTSORT:L2 Cohomology}}
[[Category:Cohomology theories]]
[[Category:Differential geometry]]
[[Category:Differential topology]]


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