Chromotopy

March 29, 2011, by Brendan Pawlowski

Way back in part 1, I gave this theorem and promised to sketch a proof:

Theorem 1. If a torus $T$ acts (smoothly) on a compact smooth manifnew $M$ with finitely many fixed points, then $\mid M^T \mid = \chi(M)$ .

If you’re willing to take this on faith, or just don’t want to think through a proof, you are allowed to skip to the discussion of an even better version of this theorem. But the proof contains some neat ideas, and in the end is just a sketch.

Proof (sketch) of theorem 1. We’ll do this by showing that the Lefschetz number of an appropriate map, homotopic to the identity, is $\mid M^T \mid$ . Any torus T has a topological generator, an element t such that the subgroup $\langle t \rangle$ generated by t is dense in T. This means that the fixed points of T are the same as the fixed points of t. The map mentioned in the first sentence is simply the action of t; certainly this is homotopic to the identity since T is connected.

We look separately at the action of T on M near fixed points and away from fixed points. Take U to be a disjoint union of coordinate patches, one around each fixed point, and $V = M \setminus M^T$ . It’s not hard to check that trace is additive, i.e. given a diagram of vector spaces with exact rows

$\begin{array}{ccccc} V_1 & \to & \cdots & \to & V_n\\ {\scriptstyle f_1} \downarrow & & \cdots & & \downarrow {\scriptstyle f_n}\\ V_1 & \to & \cdots & \to & V_n \end{array},$

$\sum_i (-1)^i tr(f_i) = 0$ . Applying this to the Mayer-Vietoris sequence for $U \cup V = M$ shows that if $f : M \to M$ maps each of $U, V, U \cap V$ into itself, then

$L(f) = L(f\mid_U) + L(f\mid_V) - L(f\mid_{U \cap V}).$

Apply this to the map t. Since U is the disjoint union of $\mid M^T\mid$ copies of a contractible space, $L(t\mid_U) = \chi(U) = \mid M^T\mid$ , so we get

$\displaystyle \chi(M) = \mid M^T\mid + L(t\mid_V) - L(t\mid_{U \cap V}).$

But $t$ has no fixed points on V, so the last two terms are 0 by the Lefschetz fixed point theorem, and we’re done!

Well, not quite. There are two issues that have been glossed over. First, $L(t\mid_U)$ doesn’t even make sense unless we know t maps U into itself, so a little more care is needed in choosing U. If p is a fixed point for the action any Lie group G, taking differentials yields a linear representation of G on the tangent space $T_p M$ . The relevant fact here is that if G is compact, this also describes the action of G in a neighborhood of p: there is a neighborhood W of p and a G-equivariant diffeomorphism $\phi : W \to T_p M$ with $\phi(p) = 0$ . One can arrange to make these neighborhoods as small as desired, so we can indeed assume that the U chosen above is T-invariant.

Second, the Lefschetz fixed point theorem had some hypotheses on the space, e.g. that it should be a compact smooth manifnew, so there is a problem applying it immediately to functions on $U, V, U \cap V$ . But these do deformation retract onto fairly obvious compact submanifnews of M, and the retractions don’t do anything to the fixed points. So that’s cool.

Let’s move on to the promised generalization of Theorem 1.

Theorem 2. If a torus T acts (smoothly) on a compact smooth manifnew M, then $M^T$ is a finite disjoint union of embedded submanifnews (perhaps of different dimensions), and $\chi(M^T) = \chi(M)$ .

To see the first assertion about $M^T$ , one uses the result stated above that near a fixed point, the action of T looks like a linear action; since the fixed point set of a linear action is a linear subspace, this gives charts realizing $M^T$ as (a union of) embedded submanifnews. The rest of the proof is analogous to the proof of theorem 1, with appropriate replacements: for example, rather than a neighborhood of a fixed point in which the action of $T$ looks linear, we need a tubular neighborhood of each component of $M^T$ in which the action looks linear.

Here’s an example. A representation $\rho : U(n+1) \to \text{GL}(\mathbb{C}^{k+1})$ of $U(n+1)$ induces an action of $U(n+1)$ on $\mathbb{CP}^k$ , and hence of the maximal torus $D \subseteq U(n+1)$ , consisting of diagonal matrices in $U(n+1)$ , on $\mathbb{CP}^k$ . For example, take V to be the standard representation of $U(2)$ (i.e. $\mathbb{C}^2$ with the usual action), and consider the representation $V \otimes V$ . Picking a basis $x, y$ of V and identifying the corresponding basis $x \otimes x, x \otimes y, y \otimes x, y \otimes y$ with the standard basis of $\mathbb{C}^4$ , the action of D on $\mathbb{CP}^3$ is

$\left(\begin{array}{cc} a & 0 \\ 0 & b \end{array}\right) [z_0, z_1, z_2, z_3] = [a^2 z_0, ab z_1, ab z_2, b^2 z_3].$

The fixed points of this D-action are $[1, 0, 0, 0], [0, 0, 0, 1]$ , and $[0, a, b, 0]$ for $[a,b] \in \mathbb{CP}^1$ . So the theorem says $\chi(\mathbb{CP}^3) = 1 + 1 + \chi(\mathbb{CP}^1)$ , and luckily $\chi(\mathbb{CP}^n) = n+1$ , so everything checks out.

More generally, by Schur’s lemma, $\rho\mid_{D}$ is the sum of 1-dimensional representations $\alpha_1, \ldots, \alpha_{n+1}$ , the weights of $\rho$ . In the example above, if $t = \text{diag}(a,b)$ , then

$\alpha_1(t) = a^2, \alpha_2(t) = \alpha_3(t) = ab, \alpha_4(t) = b^2.$

The general case will work out just like the example: a weight of multiplicity $m$ will correspond to a submanifnew $\mathbb{CP}^{m-1}$ in the fixed point set.

Unfortunately I don’t have any particular cool applications of this theorem to give, like the theorem on maximal tori last time. If nothing else, let’s show that any homogeneous space for a compact connected Lie group has nonnegative Euler characteristic; that is, if G is connected and compact, and H is a closed subgroup, then $\chi(G/H) \geq 0$ . However, we’ll only need Theorem 1.

Fix a maximal torus T in G, and consider the left action of T on G/H. It’s easy to write down the fixed points for this action. Let $K = \{g \in G : g^{-1} T g \subseteq H\}$ ; then the fixed point set is $(G/H)^T = K/H$ (this is perhaps slightly misleading notation, since K need not be a subgroup of G). The rank of a compact Lie group is the dimension of the largest torus subgroup: if $\text{rank}(H) < \text{rank}(G)$ , then we see that $(G/H)^T$ is empty. In this case, Theorem 1 implies $\chi(G/H) = 0$ .

So, suppose $\text{rank}(G) = \text{rank}(H)$ . We can assume that $T \subseteq H$ . By the theorem we proved in part 1, any two maximal tori in H are conjugate by a member of H: thus, if $g \in K$ , then there is $h \in H$ with $g^{-1} T g = h^{-1} T h$ . But this shows that $gh^{-1}$ normalizes T, so we conclude that $K = N_G(T) H$ , where $N_G(T)$ is the normalizer of T in G. Now

$K/H = (N_G(T)H) / H \simeq N_G(T) / (H \cap N_G(T))$

(these are not groups, but they’re still in bijection). Since $T \subseteq H \cap N_G(T)$ , we see that $K/H$ is naturally embedded in $N_G(T) / T$ . In the previous post I mentioned that $N_G(T) / T$ is finite (it’s the Weyl group of G), and so we see that $K/H$ is also finite (and nonempty). Therefore $\chi(G/H) = \chi(K/H) > 0$ . In fact, we get a stronger result than initially stated:

Theorem. If $\text{rank}(H) < \text{rank}(G)$ , then $\chi(G/H) = 0$ . If $\text{rank}(H) = \text{rank}(G)$ , and $W$ is the Weyl group of $G$ , then $\chi(G/H)$ is a positive integer divisor of $\mid W\mid$ .