In the theory of compact transformation groups, a basic problem reads as follows.
Let G be a compact Lie group. For a smooth manifold M, describe necessary and
sufficient conditions for the existence of a smooth action of G on a specific manifold
E (e.g., with prescribed homotopy type) such that the set Fix(G,E) of points in E
fixed under the action of G, known to be a smooth manifold, is diffeomorphic to M.
More elaborate problem reads as follows. For a smooth manifold M and a G-vector
bundle N over M, describe necessary and sufficient conditions for the existence of
a smooth action of G on a specific manifold E such that Fix(G,E) is diffeomorphic
to M and the equivariant normal bundle of M in E is stably isomorphic to N.
Among manifolds E of special interest are Euclidean spaces, disks, spheres,
projective spaces (real or complex), and spherical manifolds (e.g., lens spaces)
which are quotients of spheres by free actions of finite groups. These manifolds
E are equipped with standard (linear) actions of G, and the goal is to find how
general smooth actions may differ from the standard ones.
One also asks which compact Lie groups G act smoothly on specific manifolds E
in such a way that the set Fix(G,E) is diffeomorphic to a fixed smooth manifold M.
In the two special cases where M is empty or M is just one point, one considers
the following questions. (i) Which compact Lie groups G act smoothly on disks
(resp., Euclidean spaces) without fixed points? (ii) Which compact Lie groups G
act smoothly on spheres with just one fixed point?
If a compact Lie group G acts smoothly on specific manifolds E in such a way that
Fix(G,E) = M, a fixed smooth manifold, one asks which G-vector bundles N over M
occur stably as the G-equivariant normal bundles of M in E. In the special case where
M consists of two points and the manifolds E are spheres, one arrives on the following
question. (iii) For smooth actions of G on spheres with exactly two fixed points, which
real G-modules can occur (stably) as the tangent G-modules at the two fixed points?