Vol. 46, No. 1, 1973

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ISSN: 0030-8730
The cohomological description of a torus action

David Golber

Vol. 46 (1973), No. 1, 149–154

The theorem proved in this paper is an example of a “regularity” theorem in the study of topological group actions—that is, it shows that a general topological action of a group continues to have certain properties of “linear” actions. Consider an action of a torus T on a cohomology n-sphere X, with fixed point set the cohomology r-sphere F. Consider the map Hn(XT;Z) Hn(FT;Z), and let be the image of the generator of Hn(X;Z), considered as lying in Hnr(BT;Z), where c is an integer and η has no nontrivial integer divisors. The polynomial part η is well understood. The theorem will evaluate the integer part c in the following sense: in the linear case, c can be easily expressed in terms of the dimensions of the fixed point sets of various nonconnected subgroups of T. It is shown that this formula continues to hold in the general topological case, given some weak assumptions. There is also a corresponding result for the case F = .

Mathematical Subject Classification
Primary: 57E15
Secondary: 55B25
Received: 7 December 1971
Published: 1 May 1973
David Golber