Various physical problems formulated in euclidean or Minkowski space, or more abstractly in a quantum-mechanical Hilbert space, have properties that are not determined by local symmetries and depend on the global nature of the manifold for the theory. To understand such properties and their increasingly important role in modern physics, we must consider more formally the subjects of topology and topological spaces, differentiable manifolds, and metrics and metric spaces. Loosely, the first deals with continuity, the second with smoothness, and the third with measurement of distance. Let us now give a more detailed description of each of these, beginning with topology.

In Ch. 2 the group ${\text{S}}_3$ of permutations on three objects was used to illustrate some important group-theoretical concepts. More generally, the symmetric or permutation groups ${\text{S}}_n$ of permutations on 𝑛 objects are important in group theory for several reasons.

Non-compact groups were introduced in Ch. 12. The most important non-compact group in physics is SO(3, 1), because it is isomorphic to the group of Lorentz transformations that underlie special relativity and relativistic quantum field theory. We now investigate the Lorentz group as a non-compact group of physical interest, and as the basis for understanding spacetime symmetries and (when extended to the Poincaré group) the meaning of spin and mass for elementary particles.

Our goal in this book is to examine basic principles of symmetry, topology, and geometry in the context of modern research in physics. We begin with symmetry and the mathematical concept of a group. In this chapter some fundamental definitions and terminology will be introduced, using as illustration a few simple groups that often have transparent geometrical or combinatorial interpretations.

The rank-2 Lie algebra and associated group SU(3) were introduced in Ch. 8. Although the algebra and group structure of SU(3) can be developed without reference to physics, it is easiest to illustrate its features through concrete applications to important physical problems. In this chapter we examine the pivotal contribution of SU(3) symmetry in imposing order on the phenomenology of strongly interacting elementary particles.

Chapter 6 described how to use the technology of Clebsch–Gordan coefficients and the Wigner–Eckart theorem to calculate matrix elements for the groups SU(2) and SO(3). In this chapter we wish to extend those methods to a more complicated group and illustrate some general means for calculating matrix elements when a symmetry and associated group structure of physical interest can be attached to a problem.

In Ch. 19 we saw that the weak, electromagnetic, and strong interactions are all described by local gauge theories and, if gravity is neglected, the fundamental interactions correspond to a gauge symmetry $\text{SU}{(2)}_{\text{w}}\times \text{U}{(1)}_y\times \text{SU}{{(}_3)}_{\text{c}}$ called the Standard Model.

Chapter 2 introduced some basic concepts relevant to an understanding of group theory and its application in physics. Often these concepts have been illustrated with finite groups, although most apply with suitable modification both to finite and to continuous groups. This chapter continues our introductory survey but now the emphasis will be on continuous groups, in particular on a certain kind of continuous group called a Lie group. We have already met some examples of these groups in the preceding chapter, but now their properties will be considered more systematically.

The integer quantum Hall effect and the fractional quantum Hall effect represent extraordinary physics that appears when a strong magnetic field is applied to a low-density electron gas confined in two dimensions at very low temperature. As we shall see, these effects can be interpreted in terms of quantum numbers that are topological, and hint at the existence of whole new classes of topological matter, which will be the subject of Ch. 29.