# Simply-connected fixed point sets of smooth group actions on disks

Assume a finite group $$P$$ of prime power order acts smoothly on a disk $$D$$. We know from the Smith theory that the fixed point set $$D^P$$ is$$\mod p$$ - acyclic, where $$|P|=p^k$$ for some $$k\in\mathbb{Z}$$.

I wonder, which additional assumptions on the action of $$P$$ on $$D$$ guarntee that $$D^P$$ is simply-connected. Is it true in general? If not, maybe under some additional conditions for the action?

I know that, a piori, the conditions we assume do not guerantee even that the first homology group of $$D^p$$ (which is the abelianization of the fundamental group of $$D^P$$) vanishes - it follows from the Universal Coefficient Theorem - $$H_1(D^P)$$ may be potentially isomorphic to $$\mathbb{Z}_q$$ with $$\gcd(p,q)=1$$ for example.