The Differential Topology of Loop Spaces

Andrew Stacey

5th October 2005

Abstract

This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as background notes to a series of seminars given at NTNU and subsequently at Sheffield. The topics covered are: the smooth structure of the space of smooth loops; constructions involving vector bundles; submanifolds and tubular neighbourhoods; and a short introduction to the geometry and semi-infinite structure of loop spaces. It is meant to be readable by anyone with a good grounding in finite dimensional differential topology.

Contents
1 Introduction
2 Loopy Notation
3 Smoothly Does It
 3.1 Curvaceous Calculus
 3.2 Euclidean Loop Spaces
 3.3 The Smooth Structure of an Arbitrary Loop Space
 3.4 The Loop Space of the Tangent Space
 3.5 The Topology of the Loop Space
 3.6 Loopy Maps
 3.7 Vector Space Idol
4 Looping Bundles
 4.1 The Tangent Space
 4.2 Vector Bundles
 4.3 Principal and Gauge Bundles
 4.4 Connections
 4.5 The Enemy of my Enemy is not my Friend
5 Submanifolds and Tubular Neighbourhoods
 5.1 The Fundamental Fibration
 5.2 Tubular Neighbourhoods
 5.3 Equivariant Tubular Neighbourhoods
 5.4 A Not-So-Nice Submanifold
6 A Miscellany
 6.1 Weak Riemannian Manifolds
 6.2 More Fun with Based Loops
References