Some function spaces of CW type

Abstract

J. Milnor’s result on the CW type of certain function spaces map ( X , Y ) {\operatorname {map}}\left ( {X,Y} \right ) is extended to allow the case in which X X has a finite k k -skeleton and π i Y = 0 {\pi _i}Y = 0 , i > k i > k . One conclusion is that the self-equivalence monoid of any Postnikov stage of a finite complex has CW type. Another is that the monoid of pointed self-equivalences of a K ( π , 1 ) K\left ( {\pi ,1} \right ) manifold has contractible components when π \pi is finitely-generated.