Take a look at this:
Theorem 1. If a torus T acts on a compact smooth manifnew M with finitely many fixed points, then . Here is the fixed point set of the action, and is the Euler characteristic of M.
This is neat. Think of a circle acting freely on itself by rotation (agreeing with ), or on by rotation about a fixed axis, with the north and south pole as fixed points (agreeing with ). This immediately imposes some restrictions on possible torus actions: for example, any torus action on a manifnew of negative Euler characteristic (e.g. a multiple-holed torus) cannot have finitely many fixed points; or, by Poincaré duality, any odd-dimensional manifnew has Euler characteristic 0, so a torus action with finitely many fixed points has to act freely.
One reason to care about torus actions comes from the representation theory of compact Lie groups. From now on, G will mean a compact connected Lie group. Just as for finite groups, any finite-dimensional complex representation V of G is completely reducible, i.e. is the direct sum of irreducibles. In particular, if G is abelian, Schur’s lemma says that any irreducible representation of G is one-dimensional, so the action of G on V is diagonalizable. Even if G is not abelian, we can restrict its action on V to an abelian subgroup T to get something nice and diagonalizable, and hope this tells us something about the action of G. To get the most mileage out of this, we should choose T as large as possible. (This turns out to work quite well: for example, applying it to the adjoint representation of G on its Lie algebra, one gets the root space decomposition of the Lie algebra, and is eventually led to the classification of compact Lie groups.)
After realizing that I used T in the previous paragraph and hence that any compact connected abelian Lie group is a torus, this leads us to think about maximal tori in G: closed subgroups isomorphic to a torus and not contained in a larger torus. Such things certainly exist: the trivial subgroup is a torus, and we can only go up so far, since a containment of (closed) connected Lie groups of the same dimension is an equality.
Example. Take G to be the group of unitary matrices, U(n). The diagonal matrices in U(n) form a maximal torus: the entries along the diagonal have to be complex numbers of norm 1, and it’s easy to check that a matrix commuting with all diagonal matrices is diagonal. But there’s more to be said here: the spectral theorem shows that any unitary matrix is unitarily diagonalizable. This is the U(n) case of a more general result:
Theorem 2. a) Every element of G lies in some maximal torus. b) Any two maximal tori are conjugate.
One nice, immediate application of this theorem: the exponential map of G is surjective, because it is for a torus, and every element of G lies in a torus.
Let’s see how to prove Theorem 2 from Theorem 1. First of all, we want to be talking about fixed points, so:
Theorem 2’. a) If is in G, there is a maximal torus T such that the action (left translation) of on G/T has a fixed point. b) If S, T are maximal tori, the action of S on G/T has a fixed point.
It’s easy to write down what it means to have a fixed point in each case and see that Theorem 2’ is (a,b)-wise equivalent to Theorem 2.
We’ll need the Lefschetz fixed point theorem. Say X is a compact smooth manifnew (more generally, a retract of a finite simplicial complex), and is continuous. Write for the induced map ; this is a map from a finite-dimensional -vector space to itself. Now define the Lefschetz number of by . The Lefschetz fixed point theorem says: if L() is nonzero, then has a fixed point. (A simple fact to think of at this point: if one has a permutation of a finite set S and extends it to a linear transformation of the vector space with basis S, then the trace of is the number of fixed points of ). In particular, if is the identity, or just homotopic to the identity, then L() = χ(X).
To prove Theorem 2’, we pull a bait and switch. The fixed points of the action of a maximal torus T on G/T are easy to understand: they’re exactly the members of the Weyl group N(T)/T, where N(T) is the normalizer of T in G. In particular, there are more than 0 of them, so χ(G/T) is nonzero by Theorem 1. But now by a second appeal to Theorem 1, the action of another (maximal) torus S on G/T must also have a fixed point, which is part (b) of Theorem 2’. (One detail has been glossed over, that N(T)/T is actually finite. But this comes easily enough from thinking about how there’s a continuous map from N(T) to the discrete group .)
For part (a) of Theorem 2’, notice that the action on G/T of any in G is homotopic to the identity, simply because G is path-connected. Therefore L() = χ(G/T) is nonzero, so has a fixed point by the Lefschetz fixed point theorem. (Incidentally, we’ve shown that dim(G/T) is even, a fact which is also apparent from the root space decomposition of the Lie algebra of G.)
In part 2 I’ll sketch how to prove Theorem 1 (Lefschetz again! plus some geometry), and give a generalization to arbitrary torus actions.