A NEW LOOK AT MEANS ON TOPOLOGICAL SPACES

We use methods of algebraic topology to study when a connected topological space admits an n-mean map.


INTRODUCTION
Carath6odory and Aumann (see [1 ], [2]) were among the pioneers who first considered the question of what path-connected regions X in ]R or C could support an n-mean, that is, a map # X A" satisfying (i) z/ 1" X X, where/ is the diagonal map & X X and (ii) cr #" X X, where cr E S,, the symmetric group on n letters, acting on X by permuting components One of their main concerns was to find out if the existence of such an n-mean, n > 2, implied that X was simply connected In 1954, Beno Eckmann [4] attacked the question with the tools of algebraic topology He supposed X to be a polyhedron and only required conditions (i), (ii) above up to homotopy One of his principal conclusions was that if X is compact and admits a (homotopy) n-mean for a, ll n, then X is contractible In 1962, Eckmann, together with Tudor Ganea and the author, returned to the study of n-means in a more general setting (see [5]).Thus the n-mean defined in [4] was a morphism in the category T of based connected CW-complexes and based homotopy classes of based maps In this generality one was able to exploit the idea of mean-preserving functors.Thus if C, 79 are categories with products and F C 79 is a product-preserving functor, then F# is an n-mean in 79 for any n-mean in C Moreover, one could also examine the dual questaon of the existence of n-comeans It turns out that the concept of P-local objects and P-locahzatzon, where P s a famdy of primes, and the results related to these concepts in the categories Th and A/', the category of ndpotent groups (see [6]), enable one to mmplify many arguments m [5] and to extend the results of that paper 2. MEANS IN THE CATEGORY OF GROUPS Let be the category of groups Let n be an integer, n > 2, and let P be the famdy of primes/9 such that pn We then prove THEOREM 2.1.The group G admits an n-mean # in : (7 is commutative and P-local In that case, if we write G additively,/ is given by P HILTON (2 1) PROOF.Note first that if G is commutative, then G is P-local if and only if G admits unique division by n It is then plain that (2 1) defines an n-mean on G Conversely, let # be an n-mean on G For 9, h E G (at ths stage, we write G multiplicatively), set (9, e,..-, e) ,, (h, e,..., e) (5 Then, by condition (ii), (e, g, ., e) (e, e, ., g) ", so that, by condition (i), Similarly, h 6 But /(9, h,-.., e) 76, #(h, 9,'" ", e) 57, and #(9, h,.-., e) #(h, 9,'" ", e) Thus 7 commutes with 5, so that 9 commutes with h and G is commutative To show that G s P-local it remains to show that n th roots are unique in G. But, again using properties (i) and (ii), we conclude that #(9', e,-.., e) #(g, 9,'" ", 9) g, so that g is determined by 9 Thus G is commutative and P- local and, writing additively, we have (.qx if2, ",fin) Let G be a group and let nl >_ 2, n2 >_ 2 be ,ntegers Then G admits an n n2-mean if and only if G admits an n -mean and an n2-mean Let X be a connected CW-complex with base point.We prove, with n, P as in Section 2, THEOREM 3.1.Suppose X admits an n-mean #" X' X in Th Then X is a P-local commutative H-space PROOF.We regard the th homotopy group rr, as defining a product-preserving functor from Th to Then/.7r,# (Tr,X) rr,X is an n-mean in It follows that 7r, X s commutauve (ths s only sgmficant for 1) and P-local and that #. has the form (2 1).
Let i1" X--, X be the obxaous embedding.Then (il), IS the endomorptusm 9 !9 of the commutative P-local group 7r, X It follows that (il). is an automorphism for all i, so that/z s a self- homotopy-eqmvalence of X Let p X X be homotopy reverse to #il.Let Zl X X be the obwous embedding and let rn p/zz,2 X X.Then t s easy to see that m IS a commutauve H-structure on X We conclude that X s a P-local commutativeH-space IZI From Theorem 2.1 we deduce, more easily than in [5], TIOREM 3.2.If a compact, connected polyhedron X admits an n-mean for some n > 2, then X is contractible.
PROOF.Since the homotopy groups of X are P-local, so are the homology groups H,X, > (see [6]).Now Browder has shown [3] that a compact, connected polyhedron X which is an H-space satisfies Poincar6 duality.Thus, if X is not contractible, there exists a positive dimension N which contains the universal class giving rise to the duality isomorphism H,(X) Hv-'(X).In particular, HvX Z, but this is absurd, since Z is not divisible by n I"1 REMARK 1.We have not invoked commutativity of the H-structure in this argument If we do so, we may apply a theorem of Hubbuck showing that X would be equivalent to a product of circles, which is also impossible for a non-contractible P-local space REMARK 2. Theorem 3.2 is delicate The n-solenoid is compact and admits an n-mean but is not a polyhedron The Eilenberg-MacLane space K(Q, m) is a polyhedron and admits an n-mean for every n, but is not compact We have not proved--and doubt the truth of--the converse of Theorem 31 However, one may readily prove THEOREM 3.3.If X is a P-local, connected, commutative, associative H-space, then X admits a unique homomorphic n-mean.Further, if the connected H-space (X, m) admits a homomorphic n- mean, then (X, m) is commutative and associative.
The case n 2 admits a very neat and precise statement.If/" X X is a 2-mean on X, we define p as in the proof of Theorem 3.1 as homotopy inverse to #il, and rn p# is a commutative Hstructure on the P-local space X, where P is the family of odd primes Conversely, if m X X is a commutative H-structure on the P-local space X, we define 7-to be homotopy inverse to rnA X X (notice that rnA induces doubling on the homotopy groups of X and is therefore a self-homotopy- equivalence).Then/ 7-m is a 2-mean on X. THEOREM 3.4.The function/ p# sets up a one-one correspondence between 2-means on the P-local connected CW-complex X and commutative H-structures on X PROOF.If m p/, then z/ p# p, so % defined above, is homotopy inverse to p and 7-rn # If 7-# =/, then 7-uil so, again, p is homotopy inverse to "r and p/.t m Thus the function m 7-rn is inverse to the function 4. THE DUAL STORY Whereas the product in a familiar category (like Th, ) takes a famihar form essenlaally independent of the category, the form of the coproduct depends very much on the category m queslaon The three categories whmh will come into queslaon here are Th, , and Ab, the category of abehan groups Let C be a category adrmttmg finite coproducts, we will write C v D for the coproduct of C and D n C and Cn for the coproduct of n copies of C in C. Obwously, the symmetric group S,,, acts on C,.,, we will write V C, C for the codiagonal, which is the morphism that coincides wth the identaty on each copy of C m C, Then an n-comean on C s a morphsm for all a E S, We prove THEOREM 4.1.In PROOF.Let G be a non-trivial group and let g E G, g e If # G G, is an n-comean, n > 2, then it follows from (i) that #ge Now G, is the free product of n copies of G, so a non-trivial element of G, s umquely expressible as h,h,...h,, where G(,) ts the th copy of G in G,, h, G(,), h, :f: e, and iq iq+l, q 1,2,..-,k-1 Such an element is obviously moved under any permutation a winch moves il, so that condition (d) s violated.
I-I THEOREM 4.2.In .Ab, the abelian group A admits an n-comean, n > 2, if and only if it admits an n-mean In that case/ A A, is given by #(a)= (a a, ,a).
(41) PROOF.We note first that, in .Ab, C y D C D, so that A, A If A admits an n-mean, then, by Theorem 2.1, it is clear that (4.1) is an n-comean.Suppose conversely that # A A,, is an n- comean It is then plain from (ii) that p(a) (c, a,.-., a) for some c A such that, by (i), nc a.It remains to show that division by n is unique in A. But ,(,) (,,, r,...,,) (,, ,,..-,,), so that a is determined by na