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Concept# Loop space

Summary

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.
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 \mathcal{L}X.
As a functor, the free loop space construction is right adjoint to cartesian product with the circle, while

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We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspensions. We study long exact sequences. We construct Eilenberg-Mac Lane spaces.

We investigate the theory of principal bundles from a homotopical point of view. In the first part of the thesis, we prove a classification of principal bundles over a fixed base space, dual to the well-known classification of bundles with a fixed structure group. This leads to an adjointness property in a homotopical context between the classifying space and the loop space. We then focus on characteristic classes, which are invariants for principal bundles that take values in the cohomology of the base space. Each characteristic class captures different geo- metric features of principal bundles. We propose a uniform treatment to interpret most of known characteristic classes as obstructions to group reduction and to the extension of a universal cocycle. By plugging in the correct parameters, the method recovers several classical theorems. Afterwards, we construct a long exact sequence of abelian groups for any principal bundle. This sequence involves the cohomology of the base space and the group cohomology of the structure group. Moreover the connecting map is deeply related with the characteristic classes of the bundle.

If X is a simply connected space of finite type, then the rational homotopy groups of the based loop space of X possess the structure of a graded Lie algebra, denoted L-x. The radical of L-x, which is an important rational homotopy invariant of X, is of finite total dimension if the Lusternik-Schnirelmann category of X is finite. Let X be a simply connected space with finite Lusternik-Schnirelmann category. If dim L-x < infinity, i.e., if X is elliptic, then L-x is its own radical, and therefore the total dimension of the radical of L-x in odd degrees is less than or equal to its total dimension in even degrees (Friedlander and Halperin (1979) [8]). Felix conjectured that this inequality should hold for all simply connected spaces with finite Lusternik-Schnirelmann category. We prove Felix's conjecture in some interesting special cases, then provide a counter-example to the general case. (C) 2010 Elsevier B.V. All rights reserved.

Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space (Omega) over tildeX, that is a group. Moreover, the morphisms of topological groups (Omega) over tildeX -> G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory.