Spectral sequences and orbits of the action of groups
No. 26, 1967 (Geometry)
India ink on paper, 21.5x32 cm.
These rotating bands, almost like spectral lines themselves cast onto warped planes, mold the space that surrounds them. Indeed, they are symmetry groups, well known in complex physical problems. These groups can be discrete, or continuous. Here, the space is fibered into orbits of the action of a group, an orbit being a set of points derived from a single point on which the elements of a certain transformation group are acting, In general, each point has its own orbit, and each orbit has a certain volume and dimension. In this image, several families of orbits correspond to the actions of certain Lie groups. For example, in the lower right corner of the image, there are orbits creating concentric spheres, the result of a subgroup of a group of orthogonal transformations acting on a Euclidean space. Interesting too is the place of this object in the theory of minimal surfaces, which are invariant under the action of certain groups. If the action of the group is nice enough, then a fiber space arises whose topology can be studied with the help of spectral sequences.
