Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Chern-Weil forms and abstract homotopy theory

Authors: Daniel S. Freed and Michael J. Hopkins
Journal: Bull. Amer. Math. Soc. 50 (2013), 431-468
MSC (2010): Primary 58Axx; Secondary 53C05, 53C08, 55U35
Published electronically: April 17, 2013
MathSciNet review: 3049871
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal $ G$-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebra as the de Rham complex of a specific simplicial sheaf, and similarly give a new interpretation of the Weil model in equivariant de Rham theory. There is an appendix proving a general theorem about set-theoretic transformations of polynomial functors.

References [Enhancements On Off] (What's this?)

  • [AB] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1-28. MR 721448 (85e:58041),
  • [ABP] M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279-330. MR 0650828 (58 #31287)
  • [Bo1] A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133-150. MR 0380779 (52 #1676)
  • [Bo2] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257-281. MR 551009 (80m:55006),
  • [Br] Kenneth S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419-458. MR 0341469 (49 #6220)
  • [Bu] Ulrich Bunke, Differential cohomology, arXiv:1208.3961.
  • [C1] Henri Cartan, Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 15-27 (French). Reprinted in [GS]. MR 0042426 (13,107e)
  • [C2] Henri Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 57-71. Reprinted in [GS].
  • [ChS] Jeff Cheeger and James Simons, Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 50-80. MR 827262 (87g:53059),
  • [CS] Shiing Shen Chern and James Simons, Characteristic forms and geometric invariants, Ann. of Math. (2) 99 (1974), 48-69. MR 0353327 (50 #5811)
  • [Cu] Edward B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107-209 (1971). MR 0279808 (43 #5529)
  • [D] Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin, 1970 (French). MR 0417174 (54 #5232)
  • [DHKS] William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs, vol. 113, American Mathematical Society, Providence, RI, 2004. MR 2102294 (2005k:18027)
  • [DHZ] Johan Dupont, Richard Hain, and Steven Zucker, Regulators and characteristic classes of flat bundles, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc., Providence, RI, 2000, pp. 47-92. MR 1736876 (2001c:14042)
  • [DS] W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of Algebraic Topology, North-Holland, Amsterdam, 1995, pp. 73-126. MR 1361887 (96h:55014),
  • [E] Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 29-55 (French). MR 0042768 (13,159e)
  • [Fr] Greg Friedman, Survey article: an elementary illustrated introduction to simplicial sets, Rocky Mountain J. Math. 42 (2012), no. 2, 353-423. MR 2915498,
  • [FSS] Domenico Fiorenza, Urs Schreiber, and Jim Stasheff, Cech cocycles for differential characteristic classes - An $ \infty $-Lie theoretic construction, arXiv:1011.4735.
  • [FT] Daniel S. Freed and Constantin Teleman, Relative quantum field theory, arXiv:1212.1692.
  • [G] Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308 (98b:58156)
  • [GJ] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. MR 1711612 (2001d:55012)
  • [GS] Victor W. Guillemin and Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999. With an appendix containing two reprints by Henri Cartan [MR0042426 (13,107e); MR0042427 (13,107f)]. MR 1689252 (2001i:53140)
  • [HS] M. J. Hopkins and I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), no. 3, 329-452. MR 2192936 (2007b:53052)
  • [J] J. F. Jardine, The Verdier hypercovering theorem, Canad. Math. Bull. 55 (2012), no. 2, 319-328. MR 2957248,
  • [Joy] Dominic Joyce, D-manifolds, d-orbifolds and derived differential geometry: a detailed summary, arXiv:1208.4948.
  • [K] Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen, Math. Ann. 43 (1893), no. 1, 63-100 (German). MR 1510801,
  • [Ku] Shrawan Kumar, A remark on universal connections, Math. Ann. 260 (1982), no. 4, 453-462. MR 670193 (84d:53028),
  • [Ma] J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892 (36 #5942)
  • [Mac] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144 (96h:05207)
  • [Mi] John Milnor, The geometric realization of a semi-simplicial complex, Ann. of Math. (2) 65 (1957), 357-362. MR 0084138 (18,815d)
  • [MM] Saunders Mac Lane and Ieke Moerdijk, Sheaves in geometry and logic, Universitext, Springer-Verlag, New York, 1994. A first introduction to topos theory; Corrected reprint of the 1992 edition. MR 1300636 (96c:03119)
  • [MP] M. Mostow and J. Perchik, Notes on Gelfand-Fuks cohomology and characteristic classes (Lectures by R. Bott), Proceedings of the eleventh annual holiday symposium, New Mexico State University, 1973, pp. 1-126. reprinted in Collected Papers of Raoul Bott, vol. 3, Birkhauser, Boston, 1995, pp. 288-356.
  • [MQ] Varghese Mathai and Daniel Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), no. 1, 85-110. MR 836726 (87k:58006),
  • [MS] John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76. MR 0440554 (55 #13428)
  • [NR] M. S. Narasimhan and S. Ramanan, Existence of universal connections. II, Amer. J. Math. 85 (1963), 223-231. MR 0151923 (27 #1904)
  • [Q1] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin, 1967. MR 0223432 (36 #6480)
  • [Q2] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205-295. MR 0258031 (41 #2678)
  • [Q3] Daniel Quillen, Superconnections and the Chern character, Topology 24 (1985), no. 1, 89-95. MR 790678 (86m:58010),
  • [R] T. R. Ramadas, On the space of maps inducing isomorphic connections, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 1, viii, 263-276 (English, with French summary). MR 658951 (84h:53038)
  • [S] Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105-112. MR 0232393 (38 #718)
  • [Sch] Roger Schlafly, Universal connections, Invent. Math. 59 (1980), no. 1, 59-65. MR 575081 (81f:53022),
  • [SGA] Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.
  • [Sp] David I. Spivak, Derived smooth manifolds, Duke Math. J. 153 (2010), no. 1, 55-128. MR 2641940 (2012a:57043),

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2010): 58Axx, 53C05, 53C08, 55U35

Retrieve articles in all journals with MSC (2010): 58Axx, 53C05, 53C08, 55U35

Additional Information

Daniel S. Freed
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712

Michael J. Hopkins
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Received by editor(s): January 24, 2013
Published electronically: April 17, 2013
Additional Notes: The work of the first author was supported by the National Science Foundation under grants DMS-0603964, DMS-1207817, and DMS-1160461
The work of the second author was supported by the National Science Foundation under grants DMS-0906194, DMS-0757293, DMS-1158983
Dedicated: In memory of Dan Quillen
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society