Comptes Rendus
Géométrie et Topologie
A simple construction of the Rumin algebra
Jeffrey S. Case1
1 Department of Mathematics, Penn State University, University Park, PA 16802, USA
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1375-1382.

The Rumin algebra of a contact manifold is a contact invariant C ∞ -algebra of differential forms which computes the de Rham cohomology algebra. We recover this fact by giving a simple and explicit construction of the Rumin algebra via Markl’s formulation of the Homotopy Transfer Theorem.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.510
Classification : 53D10, 58A10, 58J10
Mots clés : Rumin complex, Rumin algebra, contact invariant
Jeffrey S. Case 1

1 Department of Mathematics, Penn State University, University Park, PA 16802, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2023__361_G8_1375_0,
     author = {Jeffrey S. Case},
     title = {A simple construction of the {Rumin} algebra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1375--1382},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.510},
     language = {en},
}
TY  - JOUR
AU  - Jeffrey S. Case
TI  - A simple construction of the Rumin algebra
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1375
EP  - 1382
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.510
LA  - en
ID  - CRMATH_2023__361_G8_1375_0
ER  - 
%0 Journal Article
%A Jeffrey S. Case
%T A simple construction of the Rumin algebra
%J Comptes Rendus. Mathématique
%D 2023
%P 1375-1382
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.510
%G en
%F CRMATH_2023__361_G8_1375_0
Jeffrey S. Case. A simple construction of the Rumin algebra. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1375-1382. doi : 10.5802/crmath.510. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.510/

[1] Robert Bryant; Michael G. Eastwood; A. Rod. Gover; Katharina Neusser Some differential complexes within and beyond parabolic geometry, Differential geometry and Tanaka theory—differential system and hypersurface theory (Advanced Studies in Pure Mathematics), Volume 82, Mathematical Society of Japan, 2019, pp. 13-40 | DOI | MR | Zbl

[2] Robert Bryant; Phillip Griffiths; Daniel Grossman Exterior differential systems and Euler-Lagrange partial differential equations, Chicago Lectures in Mathematics, University of Chicago Press, 2003, viii+213 pages | MR

[3] David M. J. Calderbank; Tammo Diemer Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math., Volume 537 (2001), pp. 67-103 | DOI | MR | Zbl

[4] Andreas Čap; Jan Slovák; Vladimír Souček Bernstein-Gelfand-Gelfand sequences, Ann. Math., Volume 154 (2001) no. 1, pp. 97-113 | DOI | MR | Zbl

[5] Jeffrey S. Case The bigraded Rumin complex via differential forms (accepted in Mem. Am. Math. Soc.) | arXiv

[6] Pierre Deligne; Phillip Griffiths; John Morgan; Dennis Sullivan Real homotopy theory of Kähler manifolds, Invent. Math., Volume 29 (1975) no. 3, pp. 245-274 | DOI | MR | Zbl

[7] Yves Félix; Stephen Halperin; Jean-Claude Thomas Rational homotopy theory, Graduate Texts in Mathematics, 205, Springer, 2001, xxxiv+535 pages | DOI | MR

[8] Pierre Julg Complexe de Rumin, suite spectrale de Forman et cohomologie L 2 des espaces symétriques de rang 1, C. R. Math. Acad. Sci. Paris, Volume 320 (1995) no. 4, pp. 451-456 | MR | Zbl

[9] Tornike Kadeishvili On the theory of homology of fiber spaces, Usp. Mat. Nauk, Volume 35 (1980) no. 3(213), pp. 183-188 International Topology Conference (Moscow State Univ., Moscow, 1979) | MR

[10] Tornike Kadeishvili Cohomology C ∞ -algebra and rational homotopy type, Algebraic topology—old and new (Banach Center Publications), Volume 85, Polish Academy of Sciences, 2009, pp. 225-240 | DOI | MR | Zbl

[11] Bernhard Keller Introduction to A-infinity algebras and modules, Homology Homotopy Appl., Volume 3 (2001) no. 1, pp. 1-35 | DOI | MR | Zbl

[12] Jean-Louis Loday; Bruno Vallette Algebraic operads, Grundlehren der Mathematischen Wissenschaften, 346, Springer, 2012, xxiv+634 pages | DOI | MR

[13] Martin Markl Transferring A ∞ (strongly homotopy associative) structures, Rend. Circ. Mat. Palermo (2006) no. 79, pp. 139-151 | MR | Zbl

[14] Michel Rumin Un complexe de formes différentielles sur les variétés de contact, C. R. Math. Acad. Sci. Paris, Volume 310 (1990) no. 6, pp. 401-404 | MR | Zbl

[15] Michel Rumin Formes différentielles sur les variétés de contact, J. Differ. Geom., Volume 39 (1994) no. 2, pp. 281-330 | MR | Zbl

[16] Michel Rumin Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal., Volume 10 (2000) no. 2, pp. 407-452 | DOI | MR | Zbl

[17] Michel Rumin An introduction to spectral and differential geometry in Carnot-Carathéodory spaces, Rend. Circ. Mat. Palermo (2005) no. 75, pp. 139-196 | MR | Zbl

[18] Dennis Sullivan Differential forms and the topology of manifolds, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) (1975), pp. 37-49 | MR

[19] Dennis Sullivan Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci. (1977) no. 47, pp. 269-331 | DOI | Numdam | MR | Zbl

[20] Chung-Jun Tsai; Li-Sheng Tseng; Shing-Tung Yau Cohomology and Hodge theory on symplectic manifolds: III, J. Differ. Geom., Volume 103 (2016) no. 1, pp. 83-143 | MR | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

La conjecture de Baum–Connes à coefficients pour le groupe Sp(n,1)

Pierre Julg

C. R. Math (2002)