PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 10, Pages 3111–3115 S 0002-9939(02)06435-3 Article electronically published on March 12, 2002 G-COINCIDENCES FOR MAPS OF HOMOTOPY SPHERES INTO CW-COMPLEXES DACIBERG L. GONC¸ALVES, JAN JAWOROWSKI, AND PEDRO L. Q. PERGHER (Communicated by Paul Goerss) Abstract. Let G be a finite group acting freely in a CW-complex Σ m which is a homotopy m-dimensional sphere and let f :Σ m → Y be a map of Σ m to a finite k-dimensional CW-complex Y . We show that if m ≥|G|k, then f has an (H, G)-coincidence for some nontrivial subgroup H of G. 1. Introduction The classical Borsuk-Ulam Theorem has been generalized in many directions; see, for example, [14], [4, Ch. 2, §34] and [10]. More recently (see [7], [8], [9], and [10]), free actions of the cyclic group Z r on S m and certain types of coincidences under maps f : S m → Y , where Y is a finite-dimensional polyhedron, were studied. It was shown that under certain dimension hypothesis there is at least one (p, r)- coincidence (see below for the definition). In the present work we study a similar problem for an arbitary finite group G which acts freely in a homotopy sphere. It turns out that results similar to those of [7], [8] and [9] remain true in this more general situation. There are many examples of finite groups, besides the cyclic ones, which act freely in homotopy spheres. Suppose that G is a finite group which acts freely in some homotopy sphere Σ m of dimension m. By [3, Ch. XVI, §9] such groups have periodic cohomology. Further, it was proved in [13, Theorem A, page 267] that if s is the period, then G acts freely on a finite simplicial complex which has the type of homotopy of a sphere of dimension ds - 1, where d is the greatest common divisor of |G| and φ(|G|), φ(|G|) being the Euler function. Thus each group which has periodic cohomology will provide an example of a free action on a homotopy sphere which is a finite CW-complex. If we do not require the complex to be finite, such groups can act in a complex of the homotopy type of a sphere of dimension s - 1 [13, Proposition 4.4, page 277]. There also exist several different actions of G in a fixed homotopy sphere. Such actions can be classified by the number of homotopy types of the orbit spaces. For more details see [6]. Received by the editors December 14, 2000 and, in revised form, May 10, 2001. 1991 Mathematics Subject Classification. Primary 55M20; Secondary 55M35. Key words and phrases. G-coincidence, G-equivariant, polyhedron, G-action, transfer, gener- alized Gysin sequence. The first author was partially supported by CNPq and FAPESP and the third author was partially supported by CNPq. c 2002 American Mathematical Society 3111 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use