Loop space

Last updated

In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A-space. That is, the multiplication is homotopy-coherently associative.

Contents

The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based loops in X, is a group, the fundamental group π1(X).

The iterated loop spaces of X are formed by applying Ω a number of times.

There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space X is the space of maps from the circle S1 to X with the compact-open topology. The free loop space of X is often denoted by ${\displaystyle {\mathcal {L}}X}$.

As a functor, the free loop space construction is right adjoint to cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory. (A related phenomenon in computer science is currying, where the cartesian product is adjoint to the hom functor.) Informally this is referred to as Eckmann–Hilton duality.

Eckmann–Hilton duality

The loop space is dual to the suspension of the same space; this duality is sometimes called Eckmann–Hilton duality. The basic observation is that

${\displaystyle [\Sigma Z,X]\approxeq [Z,\Omega X]}$

where ${\displaystyle [A,B]}$ is the set of homotopy classes of maps ${\displaystyle A\rightarrow B}$, and ${\displaystyle \Sigma A}$ is the suspension of A, and ${\displaystyle \approxeq }$ denotes the natural homeomorphism. This homeomorphism is essentially that of currying, modulo the quotients needed to convert the products to reduced products.

In general, ${\displaystyle [A,B]}$ does not have a group structure for arbitrary spaces ${\displaystyle A}$ and ${\displaystyle B}$. However, it can be shown that ${\displaystyle [\Sigma Z,X]}$ and ${\displaystyle [Z,\Omega X]}$ do have natural group structures when ${\displaystyle Z}$ and ${\displaystyle X}$ are pointed, and the aforementioned isomorphism is of those groups. [1] Thus, setting ${\displaystyle Z=S^{k-1}}$ (the ${\displaystyle k-1}$ sphere) gives the relationship

${\displaystyle \pi _{k}(X)\approxeq \pi _{k-1}(\Omega X)}$.

This follows since the homotopy group is defined as ${\displaystyle \pi _{k}(X)=[S^{k},X]}$ and the spheres can be obtained via suspensions of each-other, i.e. ${\displaystyle S^{k}=\Sigma S^{k-1}}$. [2]

Related Research Articles

In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.

In the mathematical field of topology, a section of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space, :

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and point-preserving continuous maps to the category of sets and functions. They are dual to the homotopy groups, but less studied.

In mathematics, the smash product of two pointed spaces and is the quotient of the product space X × Y under the identifications (xy0) ∼ (x0y) for all x ∈ X and y ∈ Y. The smash product is itself a pointed space, with basepoint being the equivalence class of. The smash product is usually denoted X ∧ Y or X ⨳ Y. The smash product depends on the choice of basepoints.

In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945.

In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory.

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different categories of spectra, but they all determine the same homotopy category, known as the stable homotopy category.

In mathematics, a pointed space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x0, that remains unchanged during subsequent discussion, and is kept track of during all operations.

In topology, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points. One views X as "suspended" between these end points.

In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal.

In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.

In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.

In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre, and a long coexact sequence, built from the mapping cone. Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.

In mathematics, especially in the area of topology known as algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y.

In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram for a particular concept and reversing the direction of all arrows, much as in category theory with the idea of the opposite category. A significantly deeper form argues that the fact that the dual notion of a limit is a colimit allows us to change the Eilenberg–Steenrod axioms for homology to give axioms for cohomology. It is named after Beno Eckmann and Peter Hilton.

In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem is also often stated as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction.

This is a glossary of properties and concepts in algebraic topology in mathematics.

The Whitehead product is a mathematical construction introduced in Whitehead (1941). It has been a useful tool in determining the properties of spaces. The mathematical notion of space includes every shape that exists in our 3-dimensional world such as curves, surfaces, and solid figures. Since spaces are often presented by formulas, it is usually not possible to visually determine their geometric properties. Some of these properties are connectedness, the number of holes the space has, the knottedness of the space, and so on. Spaces are then studied by assigning algebraic constructions to them. This is similar to what is done in high school analytic geometry whereby to certain curves in the plane are assigned equations. The most common algebraic constructions are groups. These are sets such that any two members of the set can be combined to yield a third member of the set. In homotopy theory, one assigns a group to each space X and positive integer p called the pth homotopy group of X. These groups have been studied extensively and give information about the properties of the space X. There are then operations among these groups which provide additional information about the spaces. This has been very important in the study of homotopy groups.

In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra of bounded linear operators on some Hilbert space H. This article describes the spectral theory of closed normal subalgebras of .

References

1. May, J. P. (1999), A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, retrieved 2016-08-27(See chapter 8, section 2)