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Proceedings of the London Mathematical Society Advance Access originally published online on February 12, 2009
Proceedings of the London Mathematical Society 2009 99(1):195-216; doi:10.1112/plms/pdn058
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© 2009 London Mathematical Society

Constructing smooth manifolds of loop spaces

Andrew Stacey

Institutt for Matematiske Fag
Norges Teknisk-Naturvitenskapelige Universitet
7491 Trondheim
Norway

Received 4 December 2006.

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite-dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists. We also show how various desired properties can be derived from the model space; for example, topological properties such as paracompactness. We pay particular attention to the fact that the loop spaces that can be defined in this way are all homotopy equivalent; and also to the action of the circle by rigid rotations.

2000 Mathematics Subject Classification 58D15.


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