One fixed point actions on low-dimensional spheres

The dimension of spheres with smooth one fixed point actions

“…On the other hand, it has been shown in M. Furuta [11, It is worthwhile remarking that the existence of a smooth one fixed point action of A 5 on S 6 implies that for any n b 7, there is a continuous one fixed point action of A 5 on S n , which is everywhere smooth, except possibly at the fixed point. This fact motivated Buchdahl-Kwasik-Schultz to show in [5,Theorem II.5] that for each n b 6, there is a locally linear, one fixed point action of A 5 on S n . Their arguments, however, do not provide ways of constructing smooth one fixed point actions on spheres.…”

Extending smooth cyclic group actions on the Poincaré homology sphere

“…In the equivariant setting we have a theorem of Cho [Cho91] on the existence of a Baire set of π-invariant metrics on X such that all the components of the fixed set M 1 (X, θ) π are either empty or smooth manifolds. This π-invariant version of Freed and Uhlenbeck is also valid on cylindrical-end 4-manifolds (it was used in [BKS90]). Even though M 1 (X, θ) π is smooth if non-empty, it may not have smooth π-invariant neighborhoods and in general the surrounding moduli space can be highly singular.…”

Untitled

“…One of Quinn's main results in [18] is an isotopy extension theorem for manifold stratified spaces, a result which is quite useful for the theory of group actions (see [26] and [2]). Quinn's methods only apply to a single isotopy at a time.…”