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THE SMOOTH STRUCTURE OF THE SPACE OF PIECEWISE-SMOOTH LOOPS

Part of: Topological manifolds Infinite-dimensional manifolds

Published online by Cambridge University Press:  10 June 2016

ANDREW STACEY
ANDREW STACEY*
Affiliation:
Oxford High School, Belbroughton Road, Oxford OX2 6XA, England e-mail: loopspace@mathforge.org
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Abstract

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We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite-dimensional manifold. We succeed for a particular type of piecewise-smooth loops. We also examine the action of the diffeomorphism group of the circle. It is not a useful action on the manifold that we define. We consider how one might fix this problem and conclude that it can only be done by completing to the space of loops of bounded variation.

MSC classification

Primary: 57N20: Topology of infinite-dimensional manifolds
Secondary: 58B05: Homotopy and topological questions
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 27 - 59
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Chen, K.-T., Iterated integrals of differential forms and loop space homology, Ann. Math. 97 (2) (1973), 217–246.Google Scholar
2. Chen, K. T., Iterated path integrals, Bull. Amer. Math. Soc. 83 (5) (1977), 831–879.Google Scholar
3. Jarchow, H., Locally convex spaces. Stuttgart, 1981. Mathematische Leitfèden. [Mathematical Textbooks].Google Scholar
4. Klingenberg, W. P. A., Riemannian geometry (Walter de Gruyter & Co., Berlin, 1995).Google Scholar
5. Kriegl, A. and Michor, P. W., The convenient setting of global analysis (American Mathematical Society, Providence, RI, 1997).Google Scholar
6. Lang, S., Differential manifolds (Springer-Verlag, New York, 1985).Google Scholar
7. Michor, P. W., Manifolds of differentiable mappings, Nantwich, Shiva Mathematics Series, vol. 3 (Shiva Publishing Ltd., Orpington, Kent, 1980).Google Scholar
8. Milnor, J., Remarks on infinite-dimensional Lie groups, in Relativity, groups and topology, II (Amsterdam, North-Holland, 1984), 1007–1057.Google Scholar
9. Omori, H., Infinite-dimensional Lie groups, Translations of Mathematical Monographs, vol. 158 (American Mathematical Society, Providence, RI, 1997), Translated from the 1979 Japanese original and revised by the author.Google Scholar
10. Pietsch, A., Nuclear locally convex spaces, Translated from the second German edition by Ruckle William H., New York, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 66.Google Scholar
11. Pressley, A. and Segal, G., Loop groups (Oxford Science Publications, New York, 1986).Google Scholar
12. Schaefer, H. H., Topological vector spaces New York, 1971. Third printing corrected, Graduate Texts in Mathematics, Vol. 3.Google Scholar
13. Seeley, R. T., Extension of C ∞ functions defined in a half space, Proc. Amer. Math. Soc. 15 (1964), 625–626.Google Scholar
14. Stacey, A., How to construct a Dirac operator in infinite dimensions. arXiv:0809.3104 [math].Google Scholar
15. Stacey, A., Constructing smooth manifolds of loop spaces, Proc. Lond. Math. Soc. 99 (1) (2009), 195–216.Google Scholar
16. Wells, J. C., Differentiable functions on Banach spaces with Lipschitz derivatives, J. Dif.Geom. 8 (1973), 135–152.Google Scholar