Rational homotopy theory

The natural setting of algebraic topology is the homotopy category. Restricting attention to simply-connected homotopy types and mappings between them allows the algebraic operation of localization (cf. Localization in categories). Inverting all the primes yields rational homotopy theory.

This theory was described algebraically by D. Quillen using differential Lie algebras modelling the loop space [a1]. It can also be described by differential algebras starting from a rational de Rham theory [a2]. A third discussion using the de Rham theory on the loop space and employing both of the first two descriptions was done by K.-T. Chen [a3].

One simple statement from all this is the following. Given a simply-connected compact manifold $M$, let $\Lambda$ be a differential graded algebra mapping into the forms on $M$ satisfying: i) $\Lambda$ consists of the real field $\mathbf R$ in degree zero and is free graded commutative in positive degrees; and ii) the mapping induces an isomorphism of real cohomology. Then: a) the homotopy groups of $M$ tensor $\mathbf R$ are naturally isomorphic to the dual of the indecomposable spaces of $\Lambda$; b) the differential yields the real form of the $k$-invariants of the Postnikov system of $M$; and c) $\Lambda$ is well defined up to an isomorphism of differential graded algebras.

The existence of $\Lambda$, mapping to the forms, the minimal model, can be shown by a simple inductive process. The above theory works almost verbatim for nilpotent spaces — those for which the fundamental group is nilpotent and the action of the fundamental group on the higher homotopy is nilpotent. The theory extends somehow to more general fundamental groups using representations coming from flat connections and minimal algebras more complicated than nilpotent ones. There are applications to Kähler manifolds [a2] and [a4].

References

[a1]	D. Quillen, "Rational homotopy theory" Ann. of Math. , 90 (1969) pp. 209–295
[a2]	D. Sullivan, "Infinitesimal computations in topology" Publ. Math. IHES , 47 (1977) pp. 269–332
[a3]	K.-T. Chen, "Iterated path integrals" Bulletin Amer. Math. Soc. , 83 (1977) pp. 831–879
[a4]	J.W. Morgan, P.A. Griffiths, "Rational homotopy theory and differential forms" , Birkhäuser (1981)