nLab Dennis Sullivan

Dennis Sullivan is an American topologist. His initial work was in geometric topology, but later he developed theories of localisation of homotopy types, rational homotopy theory, and aspects of string topology.

The editor (Andrew Ranicki) of the redistribution of his 1970 notes wrote:

The notes had a major influence on the development of both algebraic and geometric topology, pioneering

• the localization and completion of spaces in homotopy theory, including p-local, profinite and rational homotopy theory, leading to the solution of the Adams conjecture on the relationship between vector bundles and spherical fibrations,

• the formulation of the ‘Sullivan conjecture’ on the contractibility of the space of maps from the classifying space of a finite group to a finite dimensional CW complex,

• the action of the Galois group of $\overline{\mathbb{Q}}$ over $\mathbb{Q}$ on smooth manifold structures in profinite homotopy theory,

• the K-theory orientation of PL manifolds and bundles.

External links

• website

• Wikipedia entry

• interview by Kathryn Hess: video

Selected writings

On localization in homotopy theory:

• Dennis Sullivan, Geometric Topology: Localization, Periodicity, and Galois Symmetry (The 1970 MIT notes)

Introducing string topology:

• Moira Chas, Dennis Sullivan, String topology math.GT/9911159

• Ralph Cohen, John R. Klein, Dennis Sullivan, The homotopy invariance of the string topology loop product and string bracket, J. of Topology 2008 1(2):391-408; doi

Exposition of the perspective of regarding string topology-operations as the TQFT of a topological string sigma model:

• Dennis Sullivan, Sigma models and string topology, in: Mikhail Lyubich, Leon Takhtajan (eds.), Graphs and Patterns in Mathematics and Theoretical Physics, Proc. Symp. Pure Math. 73 (2005) (doi:10.1090/pspum/073, spire:1697823)

• Dennis Sullivan, Open and closed string field theory interpreted in classical algebraic topology, chapter 11 in: Ulrike Tillmann (ed.) Topology, Geometry and Quantum Field Theory, Cambridge University Press (2005) (doi:10.1017/CBO9780511526398.014)

On homotopy theory and the Adams conjecture:

• Dennis Sullivan, Genetics of homotopy theory and the Adams conjecture, Annals of Math., 100, (1974), 1–79.

On what came to be called Sullivan models:

• Dennis Sullivan, Differential forms and the topology of manifolds, in Proc. International Conf.: Manifolds Tokyo (1973), Univ. Tokyo Press (1975) 37-49 [pdf, pdf]

On rational homotopy theory:

• Dennis Sullivan, Infinitesimal computations in topology, Publications mathématiques de l’ I.H.É.S. 47 (1977) 269-331, numdam, pdf

On Sullivan models for free loop spaces:

• Micheline Vigué-Poirrier, Dennis Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976) 633-644.

On the statement that the co-binary Sullivan differential is the dual Whitehead product:

• Pierre Deligne, Phillip Griffiths, John Morgan, Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent Math (1975) 29: 245 (doi:10.1007/BF01389853)

related $n$Lab entries

• rational homotopy theory

• Sullivan conjecture

• model structure on dg-algebras

• string topology

• Simons-Sullivan structured bundle

• Whitehead product, the co-binary Sullivan differential is the dual Whitehead product

category: people