# Applications of Geometric Topology to Theoretical Physics

Geometric topology is the study of manifolds, maps between manifolds, and embeddings of manifolds in one another. Included in this sub-branch of Pure Mathematics; knot theory, homotopy, manifold theory, surgery theory, and other topics are developed in extensive detail. Do you happen to know of any applications of the techniques and/or theorems from geometric topology to theoretical physics? I'm guessing that most applications are in topological quantum field theory. Does anyone know of some specific (I'm asking for technical details) uses of say, whitney tricks, casson handles, or anything from surgery theory?

If you cannot give a full response, references to relevant literature would work as well.

• Community-wiki? – András Bátkai Dec 29 '11 at 19:36
• Have you heard about topological quantum computing? It is very closely linked to braid theory, which has given rise to a quantum algorithm for approximating the Jones polynomial. – Joe Fitzsimons Dec 29 '11 at 19:42
• @AndrásBátkai: I'm not sure I agree that this should be CW. Let's wait to see if any one has a strong opinion either way. – Joe Fitzsimons Dec 29 '11 at 19:44

In 2 dimensions, TQFTs are given by Frobenius algebras. This fact can be seen by evaluating the TQFT functor on basic building blocks of 2d manifolds: pairs of pants and discs. These give the multiplication and trace on the Frobenius algebra.

Going up in dimension, every closed 3-manifold can be obtained by a surgery of $S^3$ along a link. This allowed Reshetikhin and Turaev to define invariants of 3-manifolds with links given a modular tensor category. It turned out these invariants organize into a 3-2-1 TQFT, which gives Witten's Chern-Simons TQFT when the modular tensor category is $U_q(\mathfrak{sl}_N)$ ($q$ is a root of unity).

More generally, the proof of the cobordism hypothesis due to Lurie (classifying fully extended TQFTs) uses Morse theory to build $n$-manifolds from $(n-1)$-manifolds with handles attached (inductive construction of the category of cobordisms by generators and relations).

Similar ideas (cutting and gluing) have been applied to many areas. For example, Eliashberg et al. developed symplectic field theory, which, in particular, allows one to compute Gromov-Witten invariants of closed symplectic manifolds by reducing them to simpler objects.

• Excellent response, the links to both of those papers on arXiv will be very helpful to me and I'm interested to learn about SFT. – Samuel Reid Dec 30 '11 at 21:27

(sorry I don't have enough reputation to make a comment): This question is very broad/vague, as indeed algebraic/differential topology (symplectic geometry of course) is completely used in theoretical physics, in particular for Topological QFTs. From a physicist's perspective, start with Nakahara's Geometry, Topology, and Physics. Surgery, cobordism, and the likes are used in Knot Theory, which Witten has been studying for String Theory and other TQFT's.

And I believe the 'coolest' topic to start with is the Gauss-Bonnet Theorem, since it appears in the action functional.

• String theory is not a TQFT. Perhaps you mean topological string theory – Squark Dec 30 '11 at 11:32

there are a sequence of articles that use 4-manifold results (exotic smooth structures on R^4, casson handles, slicing knots, akbulut corks, etc) to make assertions in physics from gravity to dark energy. You can start here http://arxiv.org/abs/1112.4882 and click.

If you like Casson Handles, you'll love exotic smoothness. Basically, with an infinite tower of Casson handles one can construct a manifold which is homeomorphic but not diffeomorphic to the "usual" $\mathbb{R}^4$. The idea is to push the Morse points (which would show a topological change) "out to infinity" so that topological equivalence is maintained but there is now no diffeomorphism between the Casson handle and a subset of $\mathbb{R}^4$. Exotic $\mathbb{R}^4$, in turn, can be used to model dark matter and has numerous applications in quantum gravity.

A good physics review can be found in the text of Asselmeyer-Maluga and Brans, and a good mathematical reference is Scorpan, The Wild World of 4-Manifolds. Current research includes archive papers (most of which have been publishsed somewhere): 9610009, 1101.3168, 1112.4882.

• Funny! I just rented that book by Scorpan out from the library today and came home and saw you posted this. Looks like I'm meant to read it. :) Thanks for the post. – Samuel Reid Jan 12 '12 at 7:09

Well, I'd like to give a different perspective to the train of thought here. Geometric topology and its related fields are important in the study of elastic membranes and sheets and their d-dimensional variants. In particular, understanding topological transformations of membrane vesicles or sheets is a non-trivial and open problem. On a related note, the 1-d analogues of membranes are polymers and knot theory plays an important role in studying and classifying entanglement phenomena in dense polymer solutions. Homotopy has been more extensively used for quite some time now, in the context of classifying topological defects both in field theories and in condensed matter systems.