Math 2.0

Global analysis, analysis on manifolds Manifolds of mappings and moduli spaces Object

Free loop space

Builds on Smooth manifoldCircleLoop spaceCompact-open topologySection of a vector bundle

Description

Let MM be a smooth manifold and S1=ℝ/ℤS^1 = \R / \Z the circle. The free loop space of MM is LM=C∞(S1,M),L M = C^\infty(S^1, M), the set of all smooth loops in MM, with no condition on a base point. Since S1S^1 is compact, it is a manifold of smooth mappings, which is a Fréchet manifold. Its tangent space at a loop γ\gamma is the space of smooth vector fields along γ\gamma, that is, sections of γ∗TM\gamma^* TM.

Remarks

The circle acts on LML M by rotating the parameter, (θ⋅γ)(t)=γ(t+θ)(\theta \cdot \gamma)(t) = \gamma(t + \theta), and the fixed points of this action are the constant loops, a copy of MM. Evaluation at 00 is a map LM→ML M \to M, γ↦γ(0)\gamma \mapsto \gamma(0), whose fibre over xx is the smooth based loop space ΩxM\Omega_x M. For M=S1M = S^1, a loop has a degree in ℤ\Z, and the connected components of LS1L S^1 correspond to the integers.

In topology one also uses the space Map(S1,X)\mathrm{Map}(S^1, X) of continuous loops in a space XX with the compact-open topology. For a manifold MM, the inclusion of smooth loops into continuous loops is a homotopy equivalence.

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Circle, concept, no evidence levelCircleconceptCompact-open topology, concept, no evidence levelCompact-open topologyconceptLoop space, concept, no evidence levelLoop spaceconceptSection of a vector bundle, concept, no evidence levelSection of a vector b…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptFree loop space, object, no evidence levelFree loop spaceobject
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  1. Circle concept
  2. Compact-open topology concept
  3. Loop space concept
  4. Section of a vector bundle concept
  5. Smooth manifold concept
  6. Free loop space object

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Free_loop
About this page
TagSGGXZ3TBSA
URIm2://SGGXZ3TBSA
Kindobject
WritingW0 stub
MSC58D15, 55P35, 58E10
Instance ofManifold of smooth mappings
UsesSmooth manifold, Circle, Loop space, Compact-open topology, Section of a vector bundle

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