Tuesday, April 10, 2007


The following reading list depends heavily on Ralph Cohen's advice. Much of the material is summarized in Ralph's lecture notes written with Voronov. In particular those notes have (I hear) a good bibliography.

We will begin with Sullivan and Chas's, definition of the String operations on the loop space:

Definitions of cyclic/hoschshild homology: For cyclic we have Connes' original paper, or the Loday-Quillen
paper, which gives a modern approach.

General relation of Hochshild and cyclic homology to the homology of loop
spaces: J. Jones' paper, "Cyclic homology and equivariant homology, Inventionnes, 87 (1987), 403-423.

Minimal models: D. Sullivan, "Infinitesimal computations in topology" Publ. IHES , 47 (1977) pp. 269–331

How the hochshild cohomology under cup product, coincides with the Chas-Sullivan product under the Jones
isomorphism: Ralph's paper with Jones, "A homotopy theoretic realization of string topology", Math. Ann. 324 (2002), 773-798.

Another approach which is simpler (although only works over the reals) using differential forms and Sullivan minimal models is due to S.A. Merkulov "A deRham model for string topology"

Monday, April 9, 2007


The theme, at least for the first half of the quarter, will be the homological algebra and geometry of the loop space.

i) The String product (Sullivan and Chas)
ii) Definitions of Hoschild and cyclic cohomology of an algebra
iii) Relationship of the string product to Hoschild and cyclic cohomology on chains of loops space
iv) Relationship of the string product to the Symplectic Field Theory of the cotangent bundle