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/

Zendy Plus

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

Zendy Tools

Zendy Open

In the theory of transformation groups the smooth theory is certainly the most tractable. However, not all operations that one would like to perform remain in the smooth category. For example, analysis of the orbit space usually must go outside the smooth category. Furthermore, one would also like to study symmetries of interesting geometric spaces which often fail to be locally Euclidean such as spaces having manifolds as ramified coverings and analytic spaces. It has long been recognized that cohomology manifolds or generalized manifolds encompass all manifolds as well as many typical analytic and ramified spaces. The most important feature of generalized manifolds, from the point of view of transformation groups, is that one is often able to work with orbit spaces. In addition, one does have characteristic classes and with care one can often define workable invariant tubular neighborhoods of fixed point sets. In this paper we develop and exploit some of these ideas to prove results which even when specialized to the smooth category seem to be unobtainable from standard smooth methods. For example, we establish by cohomological methods alone formulae for the Atiyah-Singer invariant a ( ~ , (S 1, M")), Theorems 3, 4, 5, and 6. This enables us to conclude that its value is an integer and also to give explicit geometric interpretations of it in terms of the orbit space and the mapping cylinder of the orbit map. This makes it much easier to compute than by using smooth techniques alone, e.g. w 4. A very interesting geometric interpretation of ff(sl, m4k-1)=O is also given in terms of fibering M 4k-~ over a circle, w 4. There are several ingredients that enable us to obtain these results. First, we obtain by cohomological methods:

The index of manifolds with toral actions and geometric interpretations of the ?(?, (S 1,M n)) invariant of Atiyah and Singer