Rational Homotopy Theory

The second Marburger Arbeitsgemeinschaft Mathematik - MAM II will take place in Marburg from March 14th to March 18th, 2022.

It will deal with free torus actions and the so-called "Toral Rank Conjecture".  This conjecture was formulated by Steve Halperin, hence it is not surprising that it settles, among others, in the area of Rational Homotopy Theory.

Rational Homotopy Theory is a version of homotopy theory which allows for many practical computations and geometric applications. Not only do these incorporate the mentioned group actions; they range from commutative algebra to Riemannian and symplectic geometry.

As a service to anyone interested in participating in MAM II and wishing to find out more about rational homotopy, or to anyone just curious to take a very first look at the theory, we offer an introductory online mini-seminar consisting of five talks:

1. Introduction, Localization and Rational Homotopy Type, 24.11.21, 16:00. We want to show that not only the homotopy groups of a (simply connected) space can be rationalized, but the space itself. Before defining the rational homotopy type of a space, we use CW-approximation, Whitehead's theorem etc. in order to motivate the theory.

2. Commutative Differential Graded Algebras, Sullivan Models and Homotopy, 29.11.21, 14:00. We give an introduction to the algebraic aspects of the theory. For this, we describe in particular the category of commutive differential graded algebras (cdga's) and also Sullivan algebras and (commutative) differential graded modules. We find special representatives in the above category, namely Sullivan algebras as Sullivan models. In particular, we want to describe algorithmically how to construct Sullivan models for simply connected spaces explicitly. Afterwards, we define homotopy in this algebraic category, in analogy to the topological notion.

3. Lifting, 13.12.21, 14:00. We use lifting properties to prove the uniqueness of minimal models. We also show how to represent maps at the level of minimal models.

4. Spaces and Algebras: There and Back Again, 24.01.22, 14:00. At this point, we want to show how to codify topological spaces using cdga's. The key object for this is the (rational) polynomial forms functor, defined on topological spaces over the rational numbers, performing on simplices the construction of (real) differential forms on smooth manifolds. For the other direction, very roughly, cdga's can be represented as topological spaces via the Sullivan and Milnor realization. This estabishes a 1:1 correspondence between topology and algebra.

5. Fibrations and the Fundamental Theorem of Rational Homotopy Theory, 07.02.22, 14:00. In this lecture we discuss the fundamental theorem of rational homotopy theory, namely the connection between minimal models and rational homotopy groups. To this end, we deal first with the foundations, and proceed to the proof of the theorem at a second stage. We study the compatibility of fibrations with Sullivan's approach to rational homotopy theory. We bring all these notions together in order to show that the vector space underlying the minimal model corresponds to the rational homotopy groups.

If you are interested in attending these seminars, please fill the form you find below. A Zoom link will be made available to the participants.