Rational homotopy theory

Rational homotopy theory is a special idea used in mathematics, especially in a part of math called topology. It helps make some tough math problems easier by ignoring certain tricky parts called torsion in homotopy groups. This idea was started by two mathematicians, Dennis Sullivan and Daniel Quillen.

In rational homotopy theory, we can describe the basic shapes of simple spaces using special math objects called Sullivan minimal models. These models are like puzzles made with rules that help us understand the spaces better.

One cool use of this theory was in a discovery made by Sullivan and Micheline Vigué-Poirrier. They showed that for certain round, closed shapes in space, there are infinitely many different shortest paths, called closed geodesics. They used rational homotopy theory to prove this by looking at something called the Betti numbers of the space of loops on these shapes. This built on earlier work by Detlef Gromoll and Wolfgang Meyer.

Rational spaces

A rational homotopy equivalence is a special kind of map between simply connected topological spaces. It means that the map keeps important features the same when we only care about information that can be described using rational numbers.

The rational homotopy category studies these equivalences. A rational space is a type of space where all its important features are described using rational numbers. For any space, there is a related rational space that keeps the same rational features. This helps make some calculations in topology easier.

Cohomology ring and homotopy Lie algebra

In rational homotopy theory, two key features describe a space. The first is its rational cohomology ring, which is a special kind of algebra. The second is its homotopy Lie algebra, formed from the homotopy groups. These two features help mathematicians understand the space better.

Quillen showed that rational homotopy theory can be described using two different kinds of algebraic structures. One uses differential graded Lie algebras, and the other uses differential graded coalgebras. These connections make complex calculations more manageable.

Sullivan algebras

Sullivan algebras are special tools used in a part of math called rational homotopy theory. They help us understand the shape of spaces by using algebra. Imagine breaking down a complex shape into smaller, easier pieces — that’s what Sullivan algebras do!

These algebras are built from something called a graded vector space, which is just a fancy way of organizing numbers and their positions. They follow special rules that make calculations simpler, ignoring parts that don’t matter for the overall shape. This makes it easier for mathematicians to study and compare different spaces.

The Sullivan minimal model of a topological space

In rational homotopy theory, mathematicians use a special tool called the Sullivan minimal model to study shapes of spaces. For a space X, they create an algebra called A_PL(X) using polynomial forms, which helps describe the space's properties. This algebra can be very big, but it can be simplified to a smaller version called a model.

When the space X is simply connected and has certain properties, there is a unique smallest model called the Sullivan minimal model. This model helps us understand the space's rational homotopy type, which is a simpler way to study its shape by ignoring some complex details. This model connects the space's cohomology (a way to measure holes) to its homotopy groups (which describe how loops can be stretched or shrunk).

Formal spaces

In rational homotopy theory, a special kind of mathematical structure called a formal space is studied. These spaces have a property that makes their homotopy type — a way to understand their shape — completely determined by their cohomology ring. This simplifies many calculations.

Examples of formal spaces include spheres, certain types of spaces called H-spaces, symmetric spaces, and compact Kähler manifolds. However, not all spaces are formal. For example, the Heisenberg manifold is a non-formal space. Tools like Massey products help detect when a space is not formal.

Examples

Rational homotopy theory simplifies studying shapes in mathematics by ignoring certain complex details. One example is a sphere — a perfectly round shape — with an odd number of dimensions. In this theory, the sphere can be described using simple rules.

Another example involves a special kind of space called complex projective space. Even here, rational homotopy theory lets mathematicians understand its structure more easily by focusing only on what matters most. These examples show how the theory helps make tough problems more manageable.

Elliptic and hyperbolic spaces

Rational homotopy theory shows that certain spaces can be sorted into two main types: elliptic and hyperbolic. An elliptic space has a simpler structure in its rational homotopy groups, meaning these groups don't grow too quickly. Examples of elliptic spaces include spheres and some special kinds of spaces linked to Lie groups.

In contrast, hyperbolic spaces have more complex structures. For these spaces, the sizes of their rational homotopy groups grow quickly, often exponentially. This means that most finite complexes fall into the hyperbolic category. Elliptic spaces also have some neat properties, like their Euler characteristic being nonnegative.