nLab differential topology of mapping spaces

1. Introduction
2. Manifold Structure
3. Linear Mapping Spaces
4. Manifold Structure of Mapping Spaces
5. Tangent Spaces
6. The Topology of the Loop Space
7. Some Useful Smooth Maps?
8. Bundles
9. The Tangent Bundle?
10. Vector Bundles?
11. Principal and Gauge Bundles?
12. Connections?
13. The Enemy of my Enemy is not my Friend?
14. Submanifolds and Tubular Neighbourhoods
15. The Fundamental Fibration
16. Tubular Neighbourhoods
17. Equivariant Tubular Neighbourhoods
18. A Not-So-Nice Submanifold
19. A Miscellany?
20. Weak Riemannian Manifolds?
21. More Fun with Based Loops?
22. References

Introduction

This series of pages started out as an accompaniment to a series of seminars given at NTNU, and subsequently at Sheffield University, entitled:

• “The Differential Topology of the Loop Space”.

The original purpose of those seminars was to present an introduction to this topic leading up to the work contained in my preprint on the construction of a Dirac operator on loop spaces, Sta.

This was intended to be accessible to anyone with knowledge of basic finite dimensional differential topology. Thus there was a reasonable amount of background material to be explained before the subject of Dirac operators was broached. This could be divided into three broad areas: differential topology of loop spaces, spinors in arbitrary dimension, and the Atiyah-Singer index theorem (in finite dimensions).

The latter two of these topics are already superbly covered by books accessible to any differential topologist, perhaps with a little functional analysis. The book LM89 is an excellent introduction to both topics in finite dimensions whilst PR94 deals with spinors in arbitrary dimension. A group-centric viewpoint is presented in PS86, which is required reading for anyone seriously thinking about loop spaces.

The main reference for the first topic is KM97 in which is developed a theory of analysis in infinite dimensions in arbitrary topological vector spaces. However, this means that whilst being an excellent book, its subject matter is perhaps too broad and too deep for someone who just wants to know about the differential topology of loop spaces.

This led to the writing of arXiv:math.DG/0510097 with the intention was to provide a more gentle introduction than KM97 but still including the required detail to understand the differential topology of the space of smooth loops. It subsequently grew beyond that remit as more topics were added.

That article is now in the process of being imported in to the nLab. In the transfer, its scope is being extended to more general mapping spaces as many of the basic structure of loop spaces holds in this more general setting.

References

• Andreas Kriegl, Peter Michor, The convenient setting of global analysis , 1997 MR1471480

• Spinors in Hilbert space, Plymen, R. J. and Robinson, P. L., 1994 MR1312612

• Andrew Pressley, Graeme Segal, Loop groups 1986 MR900587

• Spin geometry, Lawson, H. Blaine, Jr. and Michelsohn, Marie-Louise, 1989, MR1031992