𝐻-manifolds have no nontrivial idempotents

Abstract

For a set X and a function m: XXX->X, call xEX an idempotent (element) of (X, m) if m(x, x) =x. If (X, m) is a group with unit element e, then of course e, the trivial idempotent, is the only one in (X, m). At the other extreme, if m: XXX->X is defined by m(x, x') = x, then (X, m) is a semigroup in which every element is idempotent. We remark that the set of idempotents plays an important part in both the algebraic and topological theories of semigroups. A triple (X, m, e) is an H-space if X is a topological space, eEX, and m:XXX->X is a map such that m(x, e)-m(e, x)-=x for all xEX. This generalization of the group concept is certainly of as much interest in topology as that of semigroup. One is led, therefore, to ask whether an H-space is like a topological semigroup, with its rich theory of idempotents, or like a topological group, which has no nontrivial idempotents at all. The title of the paper gives the answer that, at least when the underlying space is a manifold, an H-space behaves much like a topological group in this respect. One must, however, be careful in interpreting the statement above. For example, the interval I= [0, 1 ] is a manifold and (I, m, 0) is an H-space when m:IXI->I is defined by m(s, t)=st+|s-tI and yet (I, m, 0) has a nontrivial idempotent. Observe that if we define, for O<r<1,