Eckmann–Hilton duality

Homotopy lifting property

Homotopy extension property

Eckmann–Hilton duality is a concept in the field of algebraic topology, a branch of mathematics that uses tools from abstract algebra to study topological spaces. The duality is named after Beno Eckmann and Peter Hilton, who introduced this concept. It reveals a deep connection between two seemingly unrelated algebraic structures by showing that under certain conditions, they can be transformed into each other. This duality has significant implications for the study of homotopy theory, category theory, and the structure of topological groups.

Definition

The Eckmann–Hilton duality arises in the context of category theory and is particularly concerned with the relationship between two types of algebraic structures: loop spaces and suspension spaces. The duality states that the homotopy groups of these spaces are isomorphic under certain conditions, specifically, that the nth homotopy group of a loop space is isomorphic to the (n+1)th homotopy group of its corresponding suspension space.

Formally, if \(X\) is a pointed topological space (a space with a designated base point), then the loop space \(\Omega X\) is the space of all loops in \(X\) based at the base point, with loop composition as the operation. The suspension \(SX\) of \(X\) is a new space formed by taking the disjoint union of \(X\) with two points and then collapsing each of the two sets \(\{x,0\}\) and \(\{x,1\}\) to a point, for all \(x\) in \(X\).

The Eckmann–Hilton duality then asserts that there is an isomorphism: \[ \pi_n(\Omega X) \cong \pi_{n+1}(SX) \] for all \(n \geq 1\), where \(\pi_n\) denotes the nth homotopy group.

Implications

The implications of Eckmann–Hilton duality are far-reaching in algebraic topology and related fields. It provides a powerful tool for calculating the homotopy groups of spaces, which are fundamental invariants in topology that give information about the number of ways a space can be continuously deformed. The duality simplifies these calculations by relating the homotopy groups of loop spaces to those of their suspensions, which are often easier to compute.

Moreover, the duality has applications in the study of topological groups and their classifying spaces. It helps in understanding the structure of these groups by relating them to simpler algebraic objects through the lens of homotopy theory.

Examples

One of the classic examples of the application of Eckmann–Hilton duality is in the computation of the homotopy groups of spheres. Since the n-sphere \(S^n\) can be seen as the suspension of the (n-1)-sphere \(S^{n-1}\), the duality provides a method to relate the homotopy groups of different spheres, facilitating their calculation.

See Also

• Homotopy theory
• Algebraic topology
• Topological group
• Category theory