The Differential Topology of Loop Spaces

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.