Appendix A - The topology of manifolds

Summary

This appendix provides a quick summary of the topology needed to understand some of the more complicated constructions in (2+1)-dimensional gravity. Readers familiar with manifold topology at the level of reference or will not learn much here, although this appendix may serve as a useful reference. The approaches I present here are not rigorous: this is ‘physicists’ topology', not ‘mathematicians’ topology', and the reader who wishes to pursue these topics further would be well advised to consult more specialized sources. A good intuitive introduction to basic concepts can be found in reference, and a very nice source for the visualization of two- and three-manifolds is reference.

Mathematically inclined readers may be somewhat surprised by my choice of topics. I discuss mapping class groups, for example, but I largely ignore homology. In addition, I introduce many concepts in rather narrow settings – for instance, I define the fundamental group only for manifolds. These choices represent limits of both space and purpose: rather than giving a comprehensive overview, I have tried merely to highlight the tools that have already proven valuable in (2+1)-dimensional gravity.

Homeomorphisms and diffeomorphisms

Let us begin by recalling the meaning of ‘topology’ in our context. Two spaces M and N are homeomorphic – written as M ≈ N − if there is an invertible mapping f : M → N such that

1. f is bijective, that is, both f and f⁻¹ are one-to-one and onto; and

2. both f and f⁻¹ are continuous.