Rational Homotopy Theory

Rational homotopy theory is analogous to the study of linear algebra versus general ring and module theory. On the one hand, it is simpler, but yet has useful applications and predictive power. It isolates in algebraic topology those questions and techniques that deal with non-torsion data.

For example, if M is an orientable manifold, using the differentiable forms, one can calculate the real cohomology of M. Quillen and Sullivan realized 30 years ago that this was just the top level of information available-- in fact, information about the entire homotopy type of M is implicit in these differentiable forms. This, along with related methods of localization and completions, marked a change in the approach to the sudy of homotopy types of topological spaces.

The rational homotopy type of a space X provides the backbone, to which more detailed information concerning various primes is attached to build a picture of the homotopy type of X. The techniques involved in studying rational homotopy theory are simpler and more algebraic than those needed in traditional algebraic topology.

This course should be of use to students studying topology, commutative algebra, geometry, and several complex variables. The book by Halperin, et al provides an overall survey of the area.

@book {MR2002d:55014,

AUTHOR = {F{\'e}lix, Yves and Halperin, Stephen and Thomas, Jean-Claude},

TITLE = {Rational homotopy theory},

SERIES = {Graduate Texts in Mathematics},

VOLUME = {205},

PUBLISHER = {Springer-Verlag},

ADDRESS = {New York},

YEAR = {2001},

PAGES = {xxxiv+535},

ISBN = {0-387-95068-0},

MRCLASS = {55P62 (18Gxx 55U35)},

MRNUMBER = {2002d:55014},

MRREVIEWER = {John F. Oprea},