SUBQUOTIENTS OF A FINITE PRODUCT AND THEIR SELF-HOMOTOPY EQUIVALENCES

Given a set X = (X-1, X-2, ..., X-m) of pointed spaces, we introduce a family {X-(k,X-l)} of subquotients of X-1 x X-2 x ... x X-m. This family extends the family of subspaces of X-1 x X-2 X ... x X-m 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,X-l) is completely determined, which is used to study the group epsilon(X-(k,X-l)) of self-homotopy equivalences of X-(k,X-l). Especially, in the case of X-1 = X-2 = ... = X-m = X, we construct a homomorphism Psi((k,l)) from the semi -direct product of the m-fold product epsilon(X)(m) and the symmetric group S-m to epsilon(X-(k,X-l)) and give sufficient conditions for Psi((k,l)) to be injective. We apply this result to the case where X = S-n, CPn, or K (A(r), n) with A a subring of Q or a field Z/p, providing an important subgroup of epsilon(X-(k,X-l)).