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.

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 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. They used rational homotopy theory to prove this. This built on earlier work by Detlef Gromoll and Wolfgang Meyer.

Rational spaces

A rational homotopy equivalence is a special map between simply connected topological spaces. This map keeps important features the same when we only care about information we can describe with rational numbers.

The rational homotopy category looks at these equivalences. A rational space is a space where all its important features can be described using rational numbers. For any space, there is a related rational space that keeps the same rational features. This makes some calculations in topology simpler.

Cohomology ring and homotopy Lie algebra

In rational homotopy theory, two main ideas help describe a space. The first is its rational cohomology ring. This is a special kind of algebra. The second is its homotopy Lie algebra. This comes from the homotopy groups. Together, they help mathematicians learn more about the space.

Quillen showed that rational homotopy theory can be explained using two types of algebraic structures. One uses differential graded Lie algebras. The other uses differential graded coalgebras. These links make hard math problems easier to solve.

Sullivan algebras

Sullivan algebras are special tools used in rational homotopy theory. They help us understand the shape of spaces by using algebra.

Think of 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. This is a way of organizing numbers and their positions. They follow special rules that make calculations simpler. This helps mathematicians 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 the shapes of spaces. For a space X, they create an algebra called A_PL(X) using polynomial forms. This helps describe the space's properties.

The 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. This is a simpler way to study its shape by ignoring some complex details. The 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, we study a special kind of mathematical structure called a formal space. These spaces have a special property that makes their shape easy to understand.

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

Examples

Rational homotopy theory makes studying shapes in mathematics easier by ignoring some complicated 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. Rational homotopy theory helps mathematicians understand its structure more easily by focusing on what matters most. These examples show how the theory can make difficult problems easier to solve.

Elliptic and hyperbolic spaces

Rational homotopy theory helps us sort spaces into two main types: elliptic and hyperbolic.

An elliptic space has a simpler structure. Its rational homotopy groups do not grow too quickly. Spheres and some special spaces linked to Lie groups are examples of elliptic spaces.

Hyperbolic spaces are more complex. Their rational homotopy groups grow quickly, sometimes exponentially. Most finite complexes are hyperbolic.

Elliptic spaces have interesting properties. For example, their Euler characteristic is nonnegative.