Characteristic Fixed-Point Sets of Semifree Actions on Spheres
نویسندگان
چکیده
A group action is semifree if it is free away from its fixed-point set. P. A. Smith showed that when a finite group of order q acts semifreely on a sphere, the fixed set is a mod q homology sphere. Conversely, given a mod q homology sphere as a subset of a sphere, one may try to construct a group action on the sphere fixing the subset. The converse question was first systematically studied by Jones and then by many others. In this note, we give new numerical congruences satisfied by the homology of the fixed sets and give a definitive solution to the problem for characteristic fixed-point sets. c © 1999 John Wiley & Sons, Inc.
منابع مشابه
Generalized Rochlin Invariants of Fixed Point Sets
The Rochlin invariant for 3-manifolds is extended to higher dimensions using a result of Ochanine. Given a semifree differentiable S-action on a closed mod 2 homology sphere M with fixed point set F , generalized Rochlin invariants are definable for both M and F , and one result of this paper states that these two invariants are equal. This yields restrictions on the types of semifree different...
متن کاملSemifree Actions of Finite Groups on Homotopy Spheres
We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial. 0. Introduction. A group G is said to act semifreely on a space X if every point is e...
متن کاملDifferentiable Actions on 2 « - Spheres
Introduction. It is shown in [4] that there is an infinite family of semifree Zm actions on odd dimensional homotopy spheres. There is also an infinite family of semifree S actions on odd dimensional homotopy spheres (see [2], [5]). On the other hand, it is announced in [2] that there are only finitely many inequivalent semifree S actions on even dimensional homotopy spheres. Hence it is intere...
متن کاملDifferentiable Classification of Some Topologically Linear Actions
Let G act smoothly on S with fixed point set diffeomorphic to S O^kt^n — 3, and with all other orbits of the same type and of dimension r. Connell, Montgomery and Yang have shown that the action of G is topologically equivalent to a linear action of S if n —r^5. The problem then arises as to what can be said about the differentiable classification of these topologically linear actions. In some ...
متن کاملThe Atiyah-singer Invariant, Torsion Invariants, and Group Actions on Spheres
This paper deals with the classification of cyclic group actions on spheres using the Atiyah-Singer invariant and Reidemeister-type torsion. Our main tool is the computation of the group of relative homotopy triangulations of the product of a disk and a lens space. These results are applied to obtain lower bounds on the image of an equivariant /-homomorphism. Introduction. Smooth actions of fin...
متن کامل