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 cη be the image of the
generator of Hn(X;Z), considered as lying in Hn−r(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 = ∅.
