proceedings of the american mathematical society Volume 91, Number 3, July 1984 PRIMITIVE OBSTRUCTIONS IN THE COHOMOLOGY OF LOOPSPACES FRANK WILLIAMS Abstract. Let X and X' be //-spaces. If /: Q X -> Q X' is an //-map then the obstruction to/being a homotopy-comrnutative map is a subset {c2(/)} c [QX A QX; Q2X']. In this paper we prove: //[/] is in the image of the composition [Pk+mQX;X'} -* [LQX; X'] 3 [QX; QX'], then { c2(f)} is in the image of the composition [PkQX A P„,QX;X'] -» [LQX ALQX;X']^[QX a QX;Q2X']. Consequently if a e H"(QX; Zp) is an /f3-class in the sense of Stasheff then each • element of {c2(f)} is of the form T,c¡ ® c" where the c" are primitive. 1. The purpose of this note is to develop a decomposition formula for a certain obstruction class that occurs in the study of //-spaces. Let G and G' be associative //-spaces and /: G -* G' be an //-map. For homotopy-comrnutative G and G' we introduced in [6] the notion of / being a C2-map. Specifically, if q and q' are the commuting homotopies for G and G', respectively, and m is a homotopy from/(xy) to f(x)f(y), then/is a C2-map provided that there exists a secondary homotopy r: I2XG2^G' such that r(0, t,x,y) = f(q(t, x, y)), r(l, t, x, y) = q\t, f(x\ /(y)), r(s,0, x, y) = m(s, x, y), and r(s, 1, x, y) = m(s, y, x). The obstruction to the existence of r is an element c2(/) of [G A G; fiG'], cf. [7]. Different choices of m give us a set of obstructions {c2(/)} c [G A G; Í2G']. The sets (c2(/)} have proved to be useful in the study of //-spaces, see for example [7, 1]. We shall deal exclusively with the case G = Q,X, G' = QX', where X and X' are //-spaces and q, q' are the usual commuting homotopies for the loop multiplications. Now if G' = K(Zp, «), then {ci(f)} c [n*A ß^;ß^(Zp,«)] «ff^ßJfA nA-;Zp). Zabrodsky proved in [7] that if a g H"(tiX, Z ) is a suspension element then the elements of (c2(/)} may be written in the form £c,' ® c," where the classes c¡ and c" are also suspensions. One might hope that if a were merely a primitive element then the c'¡ and c," might also turn out to be primitive. This is in general false. In Received by the editors July 25, 1983. 1980 Mathematics Subject Classification. Primary 55P45, 55S20. Key words and phrases, //-space, homotopy-commutativity, obstruction. ©1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page 477 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use