The following problem has been posed by G. E. Bredon in his book “Introduction to compact transformation groups”, Academic Press, 1972, page 58.

Bredon’s Problem. If a compact Lie group G acts on a disk, Euclidean space, or sphere, then is it true that the fixed point set always consists of connected components of the same dimension?

For smooth actions, the answer to the question above is given in my paper “Group actions with inequivalent representations at fixed points”, Math. Zeitschrift, Vol. 187 (1984), pages 29–47. The answer depends on the quotient group G/G* of G, where G* is the connected component of the identity element of G.

Theorem. Let G be a compact Lie group. Then for any smooth action of G on an acyclic manifold (resp., a homology sphere), each connected component of the fixed point set has the same dimension if and only if G/G* does not contain an element of prime power order, i.e., G is connected or each element of G/G* is of prime power order.

The sufficiency of the restriction on G follows from the fact that if G is connected or each element of G/G* is of prime power order, then for any smooth action of G on an acyclic manifold (resp., a homology sphere with at least three fixed points), the representations of G on the tangent spaces at any two fixed points, determined via the derivation of group action, are isomorphic to each other and therefore, by the Slice Theorem, the dimension conclusion holds.

The necessity of the restriction on G is shown by proving that if G/G* contains an element not of prime power order, then there exists a smooth action of G on a disk, resp., Euclidean space; sphere, such that the fixed point set contains connected components of distinct dimensions.