Celebratio Mathematica

J. Simons and D. Sullivan: “Axiomatic characterization of ordinary differential cohomology,” J. Topol. 1 : 1 (2008), pp. 45–56. MR 2365651 Zbl 1163.57020 ArXiv math/0701077 article

The Cheeger–Simons differential characters, the Deligne cohomology in the smooth category, the Hopkins–Singer construction of ordinary differential cohomology, and the recent Harvey–Lawson constructions are each in two distinct ways abelian group extensions of known functors. In one description, these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other, they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short-exact sequences mesh with two interlocking long-exact sequences (the Bockstein sequence and the de Rham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Our first theorem shows that on the category of smooth manifolds and smooth maps, any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as mentioned earlier is unique up to a unique natural equivalence. Our second theorem shows that natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of our first theorem couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.

J. Simons and D. Sullivan: “Structured bundles define differential K-theory,” pp. 1–3 in Géométrie différentielle, physique mathématique, mathématiques et société, I: Volume en l’honneur de Jean Pierre Bourguignon [Differential geometry, mathematical physics, mathematics and society, I: Volume in honor of Jean Pierre Bourguignon]. Edited by O. Hijazi. Astérisque 321–322. Société Mathématique de France (Paris), 2008. MR 2521641 Zbl 1204.57021 incollection

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@incollection {key2521641m,
	AUTHOR = {Simons, James and Sullivan, Dennis},
	TITLE = {Structured bundles define differential
		 {K}-theory},
	BOOKTITLE = {G\'eom\'etrie diff\'erentielle, physique
		 math\'ematique, math\'ematiques et soci\'et\'e,
		 {I}: {V}olume en l'honneur de {J}ean
		 {P}ierre {B}ourguignon [Differential
		 geometry, mathematical physics, mathematics
		 and society, {I}: {V}olume in honor
		 of {J}ean {P}ierre {B}ourguignon]},
	EDITOR = {Hijazi, Oussama},
	SERIES = {Ast\'erisque},
	NUMBER = {321--322},
	PUBLISHER = {Soci\'et\'e Math\'ematique de France},
	ADDRESS = {Paris},
	YEAR = {2008},
	PAGES = {1--3},
	URL = {http://smf4.emath.fr/en/Publications/Asterisque/2008/321/html/smf_ast_321_1-3.php},
	NOTE = {MR:2521641. Zbl:1204.57021.},
	ISSN = {0303-1179},
	ISBN = {9782856292587},
}

J. Simons and D. Sullivan: “The Atiyah–Singer index theorem and Chern–Weil forms,” pp. 643–645 in Special issue: In honor of Michael Atiyah and Isadore Singer, published as Pure Appl. Math. Q. 6 : 2 (2010). MR 2761861 Zbl 1222.58015 incollection

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J. Simons and D. Sullivan: “Structured vector bundles define differential K-theory,” pp. 579–599 in Quanta of maths: Conference on noncommutative geometry in honor of Alain Connes (Paris, 29 March–6 April 2007). Edited by E. Blanchard. Clay Mathematics Proceedings 11. American Mathematical Society (Providence, RI), 2010. MR 2732065 Zbl 1216.19009 incollection

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J. Simons and D. Sullivan: “Differential characters for K-theory,” pp. 353–361 in Metric and differential geometry: The Jeff Cheeger anniversary volume (Tianjin and Beijing, China, 11–15 May 2009). Edited by X. Dai and X. Rong. Progress in Mathematics 297. Springer (Berlin), 2012. MR 3220448 Zbl 1256.19007 incollection

We describe a sequence of results that begins with the introduction of differential characters on singular cycles in the seventies motivated by the search for invariants of geometry or more generally bundles with connections. The sequence passes through an Eilenberg–Steenrod type uniqueness result for ordinary differential cohomology using these characters and a construction of a differential K-theory using Grothendieck’s construction on classes of complex bundles with connection. The last element of the sequence returns full circle with a differential character definition of differential K-theory. The cycles in this definition of characters for differential K-theory are closed smooth manifolds provided with complex structures and hermitian connections on their stable tangent bundles.

Xiaochun Rong	Related
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