Where do you need topology in physics?

[Gerenuk]

I admit I hardly know anything about topology, but I have the feeling it is a heavy, abstract part of mathematics. Yet, I heard that it can be important for physics.

In which areas concepts of topology are crucial so that results cannot be guessed by common sense alone? Where are these results essential to the physicist and not merely a prove of existence of a feature?

 

[confinement]

General point-set topology is not really that useful, because nobody has proposed any use for non-Hausdorf or other more general spaces, but point-set topology is a pre-requisite for Algebraic and Differential topology, and modern differential geometry, which have many applications in theoretical physics.

The reason that point set topology is not useful is that all of the spaces in physics are at least topological manifolds (Hausdorf space which is locally homeomorphic to R^n) or even differentiable manifolds (locally diffeomorphic to R^n) and/or metric spaces (all diff manifolds are metrizable, and so are the Hilbert spaces in QM).

One of the reasons that mathematicians began to study manifolds is because they naturally arise from differential equations. In classical mechanics, Hamilton's equations especially can be treated with differential forms on manifolds, and this allows a deeper understanding of intregrable systems (see KAM-Theorem). In chaos theory differential topology is used constantly e.g. the relation between attractor reconstruction and the embedding theorem.

Any modern understanding of GR, particle theory, or many fields of condensed matter theory, depends on understanding algebraic and differential topology.

 

[sir-pinski]

Topology is indeed a complex and abstract branch of mathematics, but it has proven to be crucial in many areas of physics. One of the main applications of topology in physics is in the study of phase transitions, which occur when a physical system undergoes a sudden change in its properties. Topological concepts, such as the study of critical points and the classification of different phases, are essential in understanding the behavior of these systems.

Another important application of topology in physics is in the study of quantum field theory. In this field, topological properties of space-time are used to describe the behavior of particles and their interactions. For example, the concept of a topological defect, such as a vortex or a soliton, is crucial in understanding the behavior of superfluids, superconductors, and other condensed matter systems.

Topology also plays a significant role in the study of cosmology, particularly in the study of the structure and evolution of the universe. The topological properties of space-time, such as its curvature and topology, are used to understand the large-scale structure of the universe and the formation of galaxies and other structures.

In all of these areas, topology provides a powerful framework for understanding complex physical systems and phenomena. It allows physicists to make predictions and draw conclusions that cannot be obtained through common sense or traditional mathematical methods. Therefore, topology is not just a proof of existence of certain features in physics, but an essential tool for understanding and explaining the behavior of physical systems.