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The answer to the last question is "no". There exist smooth actions of non-abelian compact Lie groups on euclidean space with empty fixed point set. See Bredon's book, Introduction to Compact Transfor …

The fixed point set need not be simply connected in general. If $M$ is any smooth homology $(n-2)$-sphere that bounds a smooth contractible $(n-1)$-manifold $W$ (such exist in abundance), then $S^1$ a …

There is a bit of a disconnect between the title and the actual question. Usually a semifree action is one in which the only isotropy groups are the trivial group and the whole group. The actions with …

The only oversight in your analysis is the statement that $S^1$ has only one covering with group $\mathbb{Z}$. It has only one CONNECTED covering with group $\mathbb Z$. It has plenty of others, such …

Two orientation-reversing involutions of a given closed orientable surface are equivalent if and only they have the same number of fixed point circles and have the same orientation character, in the s …

In general,the map $P(X,x_0)\to X$ from the space of based paths in $X$ is a fibration with fiber $\Omega(X,x_0)$. Since $P(X,x_0)$ is contractible, by shrinking paths back toward the base point $x_0$ …

It is a theorem of H. Hopf that a map between connected, closed, orientable n-manifolds of degree 0 is homotopic to a map that misses a point, when n > 2. See D. B. A. Epstein, The degree of a map. Pr …

Check out the paper "A Vietoris Mapping Theorem for Homotopy," by S. Smale, Proc. Amer. Math. Soc. 8 (1957), 604-610, available at http://www.jstor.org/stable/2033527 . Paraphrase of the main theore …

The simplest interesting case would be when G is Z/p, p prime. In this case the main issue is that by Smith Theory (applied to the mapping cylinder rel domain, say) you will only know that the induced …

Yes, it follows that the action of $G$ on all of $M$ is trivial. In brief this follows from what is known as "local Smith theory." Replace M by the union of $M$ and an open boundary collar on which $G …

For a counterexample take a non-simply connected homology sphere bounding a contractible manifold and remove the interior of a small ball from the contractible manifold. Such homology spheres exist in …

Yes in dimensions ≤ 2 (classical). Yes in dimension 3 via the Geometrization Conjecture (with much earlier work in special cases). No in higher dimensions, with the simplest examples perhaps being cou …

It is useful to reformulate the question in its natural differential topology setting, leaving unneeded geometric considerations aside. It is also natural to consider the analog of the problem in all …

This foundational paper: Arthur G. Wasserman, Equivariant Differential Topology, Topology, 8(1967), 127-150, has section 4 dealing with equivariant Morse theory for manifolds with a smooth action of a …