Chapter 17 - Real and Rational Homotopy Theory

Abstract

This chapter discusses the real and rational homotopy theories. It gives an exposition of the Quillan, Sullivan rational homotopy theory and the extension of this theory to real homotopy theory. The treatment is via the Sullivan approach emphasizing differential forms. To fully deal with real homotopy theory, it is essential that one uses simplicial spaces and continuous cohomology. deRham cohomology with real coefficients and carry this as far as one can without continuous cohomology. To simplify the exposition and still capture the main ideas, then shifts to rational coefficients and nilpotent simplicial sets. In this context, the chapter develops four theorems which in view form the foundation of real and rational theory.