Mathematical Field Notes

UCL Geometry and Topology Open Day talk: Floer theory

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These are the notes from the twenty minute talk I’m going to give at the UCL Geometry and Topology Open Day for prospective PhD students (November 14th 2012). For those who are interested in reading more, check out Milnor’s book on Morse theory and Atiyah’s early survey paper on Floer theory.

Given a function F\colon\mathbf{R}^n\to\mathbf{R}, the most interesting points we can study are its critical points, in other words the points where all the partial derivatives vanish.:

\frac{\partial F}{\partial x_1}=\frac{\partial F}{\partial x_2}=\cdots=\frac{\partial F}{\partial x_n}=0

For example, if F(x,y)=x^2-y^2 so that \partial F/\partial x=2x, \partial F/\partial y=-2y, then the only critical point is x=y=0. If we draw level sets of F (the contours of F considered as a height function) then the level sets undergo a significant change when we look a little below and a little above the critical height 0. This particular critical point is called a saddle point (for obvious reasons when you look at a picture of its graph).

This is the first indication of a deep relationship between the critical points of a function and the topology (or shape) of the domain of the function. Of course we could consider functions on more interesting spaces than \mathbf{R}^n. For instance, the height function on a sphere has two critical points (a maximum at the top, a minimum at the bottom) and the height function on a torus has four (a max, a min and two saddle points). You can tell what kind of critical point you have by looking at the number of downward directions of the function at the critical point: at a maximum on the torus there are two downward directions, at a saddle there is one downward direction and at a minimum there are none. To be more precise we need to recall the second derivative test which says that the Taylor expansion of a function around a critical point is dominated by its Hessian matrix. Negative eigenvalues of the Hessian mean downward directions (eigendirections) of the function, positive eigenvalues mean upward directions. Let’s define the index of a critical point to be the number of negative eigenvalues of the Hessian at that point.

The deep relationship between critical points and topology is due to Morse:

Theorem (Morse): Suppose that F\colon M\to\mathbf{R} is a function on a manifold M and that, for all critical points, the Hessian matrix has no zero eigenvalues (nondegeneracy). Then there are at least b_i critical points with index i where b_i is a purely topological quantity called the ith Betti number, counting the number of “linearly independent i-dimensional holes”.

For instance, in a torus there are two “linearly independent 1-dimensional holes” and in our example we had two critical points of index 1 (saddles). Conversely, with a little more information about the gradient flowlines of the function, one can say something about the topology of M from data about the function F and this is a very fruitful piece of topology which you can read about in Milnor’s classic textbook on Morse theory.

More interestingly, one can study functions on infinite-dimensional spaces M. For example, M might be the space of paths \gamma\colon[0,1]\to K in a Riemannian manifold K with fixed endpoints and the function F might be the length functional

F(\gamma)=\int_0^1|\dot{\gamma}(t)|dt

The critical points of this functional are familiar: they are the solutions to the Euler-Lagrange equation, in other words the geodesic paths. In this case you can still make sense of the index of a critical point – though M is infinite-dimensional, there are only a finite-dimensional space of downward directions. Indeed when \gamma is length-minimising there are no downward directions! Using this kind of Morse theory on the loopspaces of certain homogeneous spaces, Bott was able to prove his remarkable periodicity theorem for the homotopy groups of the orthogonal and unitary groups.

Yet more interesting is the case when the index becomes infinite, the case when your critical points have infinitely many upward and infinitely many downward directions. This is called Floer theory or semi-infinite Morse theory, and is significantly harder. Let me give you an example.

Let H\colon\mathbf{R}^3\to\mathbf{R} be a function. On the space of 2\pi-periodic functions \mathbf{R}\to\mathbf{R}^2, in coordinates t\mapsto (p(t),q(t)), you can define the symplectic action functional

(p,q)\mapsto\int_0^{2\pi}(p(t)\dot{q}(t)-H(t,p(t),q(t)))dt

The Euler-Lagrange equations are

\frac{\partial L}{\partial p}=\frac{d}{dt}\frac{\partial L}{\partial\dot{p}},\qquad\frac{\partial L}{\partial q}=\frac{d}{dt}\frac{\partial L}{\partial\dot{q}}

which become

\dot{q}=\frac{\partial H}{\partial p},\qquad\dot{p}=-\frac{\partial H}{\partial q}

These you may recognise as Hamilton’s equations of motion for the Hamiltonian function H. So a critical point of the symplectic action functional is a 2\pi-periodic orbit of this Hamiltonian system. Let me convince you that it has infinite index, at least when H\equiv 0. Let p(t)=\sum_{k=-\infty}^{\infty}p_k^{ikt} and q(t)=\sum_{k=-\infty}^{\infty}q_ke^{ikt} where p_{-k}=\bar{p}_k and q_{-k}=\bar{q}_k are the Fourier coefficients. Then

F(p+\delta,q+\epsilon)=\sum_k\sum_{\ell}(p_k+\delta_k)i\ell(q_\ell+\epsilon_{\ell})\int_0^{2\pi}e^{i(k+\ell)t}dt

The integral is nonzero only when k=-\ell, when it is \pi, so this becomes

F(p+\delta,q+\epsilon)=\sum_{k=-\infty}^{\infty}k\pi(p_k+\delta_k)(q_k+\epsilon_k)

This is

F(p+\delta,q+\epsilon)=\pi\sum_{k=-\infty}^{\infty}(kp_kq_k+k(p_k\epsilon_k+q_k\delta_k)+k\delta_k\epsilon_k)

The Hessian is (twice) the quadratic part of this. If we define

\mathbf{H}=\pi\left(\begin{array}{ccccccc}  \ddots & 0 & 0 & 0 & 0 & 0 & 0\\  0 & -2 &0 &0 & 0 & 0 & 0 \\  0 & 0 & -1&0 &0 & 0 & 0 \\  0 & 0 & 0 & 0 &0 &0 & 0 \\  0 & 0 & 0 & 0 &1 &0 & 0 \\  0 & 0 & 0 & 0 &0 &2 & 0 \\  0 & 0 & 0 & 0 &0 &0 & \ddots  \end{array}\right)

then the Hessian is

\left(\begin{array}{cc}\delta &\epsilon\end{array}\right)\left(\begin{array}{cc}0 & \mathbf{H}\\ \mathbf{H} & 0\end{array}\right)\left(\begin{array}{c}\delta\\ \epsilon\end{array}\right)

and we see that the Hessian is an infinite matrix with infinitely many positive and negative eigenvalues!

Floer’s idea was to prove some kind of invariance of this theory under deformations of H, which allowed him to compare the number of critical points (periodic orbits) for an arbitrary H to the number of periodic orbits for a particular H he could understand. He used this to prove the Arnold conjecture: a highly nontrivial lower bound on the number of periodic orbits.

Morse theory is related by Morse’s theorem to the topology of a manifold. One exciting open problem is to understand the topology of “semi-infinite cycles” and their intersection theory, which is what Floer’s theory seems to capture. A slighty more precise version of this open problem goes by the name of the Atiyah-Floer conjecture. You can read about it here:

M. F. Atiyah, “New invariants of 3- and 4-dimensional manifolds” in The Mathematical Heritage of Hermann Weyl (Proceedings of Symposia in Pure Mathematics, Volume 48) 1988 p. 285 – 300

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Written by Jonny Evans

November 13, 2012 at 9:37 pm

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