# Real world uses of homotopy theory

I covered homotopy theory in a recent maths course. However I was never presented with any reasons as to why (or even if) it is useful.

Is there any good examples of its use outside academia?

Homotopy theory / algebraic topology was born out of applications rather than abstract nonsense considerations. So there's plenty of applications, as that's how the subject began.

Perhaps the first topological proof would be the bridges of Konigsberg problem: http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg

Where algebraic topology started getting off the ground was in the work of Poincare. The Poincare-Hopf Index theorem: http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem

was a landmark. In its natural setting it was a relationship between Euler characteristic, tangent bundles and intersection theory. But from the perspective of a differential equator it's a fundamental tool that allows you to determine whether or not differential equations have fixed points.

Applications have piled-up over the years. Some of the more modern ones are listed in other people's responses. The birth of topological dynamics in the mid 20-th century was of course a big one.

Robert Ghrist is an amazing applied mathematician who uses a lot of interesting algebraic topology for engineering applications. He uses homology and sheaf theory. I claim this answers your question since homology is a generalization of homotopy theory.

• What you wrote isn't even true. Homology is not a generalization of homotopy theory.
– user126
Jul 24 '10 at 21:24
• @irritating-string-of-numbers You caught me, in your purposely poorly posed(accidental alliteration) question, you tricked me into thinking this hypothetical homotopy neophyte was talking about pi_1. In that case, simplicial homology is a generalization, in the sense that it detects n-holes. However, in the strict sense, homology is bigger than just detecting holes(see the other question on this), and homotopy theory is much bigger than pi_1. In fact, in my experience, you know much more than I about homotopy theory. I apologize for not being careful. Jul 24 '10 at 21:33
• If you're working with spectra, then homology is by definition a generalization of homotopy. (Since this is stable, it doesn't explicitly include $\pi_1$ in all its nonabelian glory, of course.) Is this in the 'other question on this', or is that about something else? Nov 5 '10 at 6:29

Homotopy groups and homotopy classes of maps are used in physics to study topological defects. Roughly speaking, $\pi_k(X)$ classifies codimension $k+1$ defects in textures modeled on $X$ (maps from $R^d$ to $X$ continuous except at the loci of defects). Other homotopy classes of maps can be used to study linked defects.

See the following beautiful review paper by N.D. Mermin for an introduction: http://rmp.aps.org/abstract/RMP/v51/i3/p591_1

Homotopy theory is useful in quantum mechanics when (for instance) talking about manifolds of Hamiltonians. You might have a collection of Hamiltonians that depend continuously on some parameters $p_1, p_2, \ldots, p_n$ such that the matrix representation of the Hamiltonian is periodic in some subset of the parameters $p_{r_1}, p_{r_2}, \ldots, p_{r_m}$. From this, the fundamental group of the manifold of Hamiltonians can be computed, which has some physical ramifications.

A concrete example of this would be the Hamiltonian that describes the quantum hall effect. The quantum hall effect is the phenomena that the resistance in a 2-dimensional substrate exposed to a perpendicular electric field at close to zero temperature is quantized. In condensed matter physics there is a notion called quasimomentum that can be thought of as being related to momentum but is a bit different. We need something different from the classical definition of momentum because the classical definition depends on translation invariance, and in a crystal there is only discrete translation invariance. In the quantum hall effect, we have two quasimomenta: $k_x, and k_y$, corresponding to quasimomentum in the $x$ and $y$ directions. The Hamiltonian is periodic in both of these parameters, leading to a fundamental group of the manifold of Hamiltonians of $\mathbb{Z}^2$, i.e. the fundamental group of the torus $T^2$.

There's a lot more to the topic than this. This paper by Avron, Seiler, and Simon has more details:

Homotopy and Quantization in Condensed Matter Physics

• Just because something invokes the fundamental group does not mean that it involves homotopy theory...
– user126
Aug 5 '10 at 11:44
• @97832123: That's a rather dubious statement. As you probably know, all the higher homotopy groups are $\pi_1$'s of iterated loop spaces. Aug 6 '10 at 21:04
• @Pete: Would you say that any theorem involving the Nullstellensatz automatically involved algebraic geometry?
– user126
Aug 8 '10 at 19:24
• I think I'm with 978 on this one. I'm often disappointed when I see that geometric topologists, symplectic topologists, etc. don't use homotopy theory beyond throwing around a few $\pi_1$-actions. Homotopy theory has so much more to say! But I guess maybe it's not always sufficiently obviously related to what they're thinking about. Still, it's a bit like how of all Beethoven's symphonies, most people only know Ode to Joy and the first 8 notes of 5th. Nov 6 '10 at 18:05

Homotopy methods can be used for solving nonlinear equations (algebraic and differentials) where these equation appear in different problems in engineering and science. One example of these equations is a system of nonlinear algebraic equations which is a model of electrical circuit. This mathematical model consists of two equations which can be easily solved using Newton homotopy to find the value of the current in this circuit. For more information, please see my PhD thesis entitled "Newton homotopy algorithms for solving nonlinear systems" and my papaers presented in some conferences and published in some journals upon my name: Talib hashim Hasan.

In this paper http://www.ncbi.nlm.nih.gov/pubmed/18485361 people started to classify RNA structures looking at some kind of topological genus.