Celebratio Mathematica — Simons

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Works connected to Jeff Cheeger

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J. Cheeger and J. Simons: “Differential characters and geometric invariants,” pp. 50–80 in Geometry and topology (College Park, MD, 1983–1984). Edited by J. Alexander and J. Harer. Lecture Notes in Mathematics 1167. Springer (Berlin), 1985. MR 827262 Zbl 0621.57010 incollection

This paper first appeared in a collection of lecture notes which were distributed at the A.M.S. Summer Institute on Differential Geometry, held at Stanford in 1973. Since then it has been (and remains) the authors’ intention to make available a more detailed version. But, in the mean time, we continued to receive requests for the original notes. Moreover, the secondary invariants we discussed have recently arisen in some new contexts, e.g., in physics and in the work of Cheeger and Gromov on “collapse” (which was the subject of the first author’s lectures at the Special Year). For these reasons we decided to finally publish the notes, albeit in their original form.

Jeff Cheeger	Related
James W. Alexander	Related
John L. Harer	Related

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
Dennis P. Sullivan	Related
Xianzhe Dai	Related
Jeff Cheeger	Related