MEMORIA DESCRIPTIVA ESTRUCTURALES

We were unable to present the full theory of the corank 2 orbits in this thesis. But a sketch of our approach is as follows:

1. To enumerate the possible groups which are corank 2 insideS1_×E 6.

2. Using the list in (1) and Theorem 7.16 to find those groups with representations on which they act with codimension one and such that the generic orbit of

such an action is stabilized by the groupS1×S1×SU(4). This is a neccesary requirement for any such orbit.

3. The use of Theorem 8.6 in [Bredon, 1972, pg. 211] on the action of a compact Lie group on a compact manifold with codimension 2 to prove that the orbit spaceS1×E6\E7/S3×Spin(12) admits a description as a polytope.

4. We identify 2 of the vertices of this polytope with the full rank orbits and the interior with the generic orbit. Lemma 9.13 shows that corank one orbits cannot be vertices and so any remaining vertices must be of corank 2.

5. We show using a cohomological argument on dimension that H2(E7/S3 ×

Spin(12)) is isomorphic to the disjoint union of the second cohomology groups of the orbits which are both vertices on the polytope and corank 2. We use this to show there is exactly one corank 2 vertex orbit with stabilizer S1 ×

S3×Spin(7).

6. We analyse the decomposition of these actions in the sphere bundle to the normal bundle at the vertex bundle to obtain the orbit type of the edges of the polytope

7. We conclude with a complete description of the decomposition as the homotopy colimit of a diagram obtained as a barycentric subdivision of the 2-simplex.

Full details are not given of this method as while I have completed many of the computations I have not been able to describe the argument in the time I had to finish this thesis.

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