PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 3, March 1988 THE SET OF BALANCED POINTS WITH RESPECT TO S1 AND S3 ACTIONS OF MAPS INTO BANACH SPACE NE2A MRAMOR-KOSTA (Communicated by Haynes R. Miller) ABSTRACT. Let G be the group of units in the field F, which is either R, C or H, let X be a free G-space, and let / be a map from X to a Banach space E over F. In this paper we give an estimate for the size of the subset of X consisting of points at which the average of / is equal to zero. The result represents an extension of the Borsuk-Ulam-Yang theorem. Let F be one of the fields R, C or H, and G C F the unit sphere equipped with the standard group structure. Take any space A with a free action of G, let V be a finite-dimensional vector space over F with the standard action of G (multiplication by units), and let /: A —► V be a continuous map. The size of the set of points in X at which the average of / is zero can be described in terms of an invariant called the index [5]. We would like to show that for a certain class of maps, this can be generalized to the case where the representation space V is replaced by an arbitrary Banach space E over F. This has already been done by Spannier and Holm [4] for the field R, that is, if A is a space with a free action of G = Z2. The average of a map / : X —► E, from a space A with a free action of G to a representation space F for G (possibly infinite-dimensional), is the map Av/: X —► E, defined by Av f(x) — fG g"1 f(gx) dg, where / denotes the Haar integral on G. The point ¡r € A is a balanced point of / if Av f(x) = 0. The set of balanced points is denoted by A(f). Proposition l. Let f: X -* F be continuous. (a) Av/ is an equivariant continuous map. (b) /// is equivariant, then Av/ = /. (c) For any map f, A(f) is a closed invariant subset of X and A(f) = A(Av/). PROOF, (a) Av/ is equivariant, since for any h EG, Avf(hx)= f g-1f(ghx)dg = h f (gh)'1 f(ghx)d(gh) = hkvf(x). JG JG The map F: G x X —> E, defined by F(g,x) = f(gx), is continuous. Thus, for any £ > 0 and for any pair (g,x) G G x A, there exist open neighborhoods U9tX of g and Vg,x of x such that \\f(gx) —f(hy)\\ < £ for any h G Ug,x and y e Vg,x. Received by the editors November 24, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55M20, 55M40; Secondary 47H09. Key words and phrases. Average, equivariant map, compact map, characteristic class, index, coindex of equivariant map. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 723 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use