English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A linear action of an abelian group on a sphere generally contains a large family of invariant linear subspheres. In this paper the problem of finding invariant subspheres for more general smooth actions on homotopy spheres is considered. Classification schemes for actions with invariant subspheres are obtained; these are formally parallel to the classifications discussed in the preceding paper of this series. The realizability of a given smooth action as an invariant codimension two sub- sphere is shown to depend only on the ambient differential structure and an isotopy invariant. Applications of these results to specific cases are given; for example, it is shown that every exotic 10-sphere admits a smooth circle action. In our previous papers in this series (45,44), we have considered the theory of semifree actions on homotopy spheres as formulated by W. Browder and T. Petrie (10) and M. Rothenberg and J. Sondow (34). Specifically, in the first paper a method was presented for describing (at least formally) those exotic spheres admitting such semifree actions- a problem first posed explicitly by Browder in (3, Problem 1, p. 7) -and the seconid paper extended the whole theory to handle certain actions that are not semifree. This paper will treat another problem posed in Browder's paper (3, Problem 3) regarding invariant subspheres of homotopy spheres with group actions. One motivation for considering this question is that linear actions on spheres generally admit a great assortment of invariant linear subspheres (e.g., if the group is abelian and the dimension is much larger than the group's order), and from this viewpoint the existence of invariant subspheres reflects the extent to which an arbitrary smooth action resembles some natural linear model. In particular, this idea is central to the work of Browder and Livesay on free involutions (8) (compare also (3)). The existence of such subspheres is directly related to the realizability of actions as equivariant smooth suspensions, providing basic necessary conditions for such a realization. We shall also consider a dual problem in this paper; namely, the description of those group actions that can be smoothly equivariantly suspended. Questions of this sort first arose in the study of free involutions (16), and their close

Differentiable group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions