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# Lie groups and homogeneous spaces

In the Fifties Bott applied Morse theory with great success to the topology of Lie groups and homogeneous spaces. In [8] he showed how the diagram of a compact semisimple connected and simply connected group determines the integral homology of both the loop space and the flag manifold , where is a maximal torus.

Indeed, Morse theory gives a beautiful CW cell structure on , up to homotopy equivalence. To explain this, recall that the adjoint action of the group on its Lie algebra restricts to an action of the maximal torus on . As a representation of the torus , the Lie algebra decomposes into a direct sum of irreducible representations

where is the Lie algebra of and each is a -dimensional space on which acts as a rotation , corresponding to the root on . The diagram of is the family of parallel hyperplanes in where some root is integral. A hyperplane that is the zero set of a root is called a root plane.

For example, for the group and maximal torus

the roots are , and the diagram is the collection of lines in the plane in as in Figure 3. In this figure, the root planes are the thickened lines.

For and

the Lie algebra is , the roots are , and the adjoint representation of of on corresponds to rotations. The root plane is the origin.

A point in is regular if its normalizer has minimal possible dimension, or equivalently, if its normalizer is . It is well known that a point in is regular if and only if it does not lie on any of the hyperplanes of the diagram. If is regular, then the stabilizer of under the adjoint action of is and so the orbit through is .

Choose another regular point in and define the function on to be the distance from ; here the distance is measured with respect to the Killing form on . Let be all the points in obtained from by reflecting about the root planes. Then Bott's theorem asserts that is a Morse function on whose critical points are precisely all the 's. Moreover, the index of a critical point is twice the number of times that the line segment from to intersects the root planes. This cell decomposition of Morse theory fits in with the more group-theoretic Bruhat decomposition.

For and the set of diagonal matrices in , the orbit is the complex flag manifold , consisting of all flags

Bott's recipe gives 6 critical points of index respectively on (See Figure 5). By the lacunary principle, the Morse function is perfect. Hence, the flag manifold has the homotopy type of a CW complex with one 0 -cell, two -cells, two -cells, and one -cell. Its Poincaré polynomial is therefore

Next: Index of a closed Up: The life and works Previous: Morse theory
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