SUBQUOTIENTS OF A FINITE PRODUCT AND THEIR SELF-HOMOTOPY EQUIVALENCES

Abstract

Given a set X = (X1, X2, . . . , Xm) of pointed spaces, we introduce a family {X(k,l)} of subquotients of X1 × X2 × · · · × Xm . This family extends the family of subspaces of X1 × X2 × · · · × Xm introduced by G. J. Porter and contains the product, the fat wedge, the wedge and the smash product. The (co)homology with field coefficients of X(k,l) is completely determined, which is used to study the group E(X(k,l)) of self-homotopy equivalences of X(k,l). Especially, in the case of X1 = X2 = · · · = Xm = X , we construct a homomorphism (k,l) from the semi-direct product of the m-fold product E(X)m and the symmetric group Sm to E(X(k,l)) and give sufficient conditions for (k,l) to be injective. We apply this result to the case where X = Sn , CPn , or K (Ar , n) with A a subring of Q or a field Z/p, providing an important subgroup of E(X(k,l)).