Nonlinear analogs of linear group actions on spheres

Group actions on homology spheres

This can be stated in a more symmetric manner. Let r be any positive integer not equal to 3. Then n acts freely and homologically trivially on Z r i ff n acts freely and homologically trivially on SL In fact, there is a one-to-one correspondence between such actions on U and such actions on S r. (The classification of such actions is discussed in w In addition the actions constructed have the p...

Pseudofree Group Actions on Spheres

R. S. Kulkarni showed that a finite group acting pseudofreely, but not freely, preserving orientation, on an even-dimensional sphere (or suitable sphere-like space) is either a periodic group acting semifreely with two fixed points, a dihedral group acting with three singular orbits, or one of the polyhedral groups, occurring only in dimension 2. It is shown here that the dihedral group does no...

Linear and Nonlinear Group Actions , and The

One of the simplest and most useful notions in mathematics is that of a group action: if G is a group and X is a nonempty set, then an action of G o n X or a G-set structure on X consists of a multiplication operation G X ! X, with the image of a pair g;x written as, say, gx, with the following axioms satissed: 1 1x = x for all x 2 X here 1 2 G is the identity element o f G; 2 ghx = ghx for all...

Some Cyclic Group Actions on Homotopy Spheres

In [4J Orlik defined a free cyclic group action on a homotopy sphere constructed as a Brieskorn manifold and proved the following theorem: THEOREM. Every odd-dimensional homotopy sphere that bounds a para-llelizable manifold admits a free Zp-action for each prime p. On the other hand, it was shown ([3J) that there exists a free Zp-action on a 2n-1 dimensional homotopy sphere so that its orbit s...

A Note on Group Actions on Homology Spheres