PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 1, May 1974 A COINCIDENCE THEOREM RELATED TO THE BORSUK-ULAM THEOREM FRED COHEN AND J. E. CONNETT Abstract. A coincidence theorem generalizing the classical result of Borsuk on maps of S" into Rn is proved, in which the anti- podal map is replaced by a Z„-action on a space which is (n — l)(p — l)-connected. The main result is: Theorem 1. Let X be a Hausdorff space which supports a free ZP-action, andf.X^-R7' a continuous map, n^.2. If X is (n —l)(p—l)-connected, then there exists x e X and g e Zv, g^identity, such that f(x)=f(gx). We observe that if/» = 2, then Theorem 1 is a restatement of the classical Borsuk-Ulam theorem. The case «=2 has been studied by the second author [3], using the fact that Artin's braid groups have no elements of finite order. For this case it suffices to assume only that ttx(X) is a torsion group. The cases «>2 require a bit more geometry. We recall the definition of the configuration space F(M,j), of j distinct points in a space M: F(M,j) is the subspace of M> given by {(xx, ■■■ , Xj)\xt e M, x^x,- if zV/'}- The spaces F(M,j) have been studied by Fadell and Neuwirth [4]. Evidently .Sy, the symmetric group on y letters, acts freely on F(M,j) by permutation of coordinates. We define F(RCD,j) to be inj limn F(Rn,j), where F(Rn,j)<=F(Rn+1,j) is given by the standard inclusion of Rn in RB+1. By [2], F(R°°,y) is contractible. Since Z„, the cyclic group of order/», acts on F(Rn,p) and F(Rca,p) via the action given by a homomorphism ZP-+LP which sends 1 e Zp to the cycle (1, 2, ■••,/»), it follows that F(iv°°,p)\Zp is a A:(Z„, 1)- space. We shall assume without loss of generality that p in the hypothesis of Theorem 1 is prime. With these preliminaries, we state the main lemma; the lemma's proof is deferred till after the proof of Theorem 1. Lemma 2. H\F(Rn,p)\Zp;Zv)=0 ///>(//-!)(/»-!). Received by the editors August 8, 1973. AMS (MOS) subject classifications (1970).Primary 55C20, 55C35. © American Mathematical Society 1974 218 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use