The answer to the question in the Bredon Problem is given in my paper ,,*Group actions with inequivalent representations at fixed points*“, Math. Zeitschrift, Vol. 187(1984), pages 29–47.

It turns out that for a compact Lie group *G*, 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 such that 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 and for any two fixed points x and y, the G-representations determined on the tangent spaces at x and y are isomorphic to each other. The same conclusion holds for any smooth action of G on a homology sphere with at least three fixed points.*

**Corollary.** *Let G be a compact Lie group such that G is connected or each element of G/G* is of prime power order. Then, for any smooth action of G on any acyclic manifold (resp., homology sphere), any two connected componets of the fixed point set F are of the same dimension. *

**Theorem. ***Let G be a compact Lie group such that G/G* has an element not of prime power order. Then there exists a smooth action of G on some Euclidean space (resp., disk; sphere) such that the fixed point set F has two connected components of distinct dimensions.*

**Corollary. ***Let G be a compact Lie group. Then, for every smooth action of G on some Euclidean space (resp., disk; sphere), the connected components of the fixed point set F all are of the same dimension if and only if each element of G/G* is of prime power order. *

The following problem goes back G. E. Bredon, “Introduction to compact transformation groups”, Academic Press, 1972, page 58.

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