Notes on proper homotopy theories associated with complact $PL$_manifolds

E. Domínguez, L. J. Hernández
1982 Publicacions matemàtiques  
The purpose of this note is to present a bigraded sequence of functors associated with compact PL-manifolds . These functors are invariant< of proper homotopy type, and distinguish topological spaces which liave the same homotopy type . On the category of compact topological spaces thesifunctors agree with the llurewicz homotopy groups, hence the principal interest of these functors is for noncompact spaces . Consider the category of compact PL-manifolds and the bordism theory between these
more » ... folds . We are thinking of closed PL-manifolds M of dimension n-m, PL-imbedded into Sn (M can be the empty set), apd we cai ask about the different ways that Sn -M can be mapped by a proper map inu a topological space . We will consider a proper map as a continuous maT f :X -> Y such that for any compact closed subset K of Y, f _1 (K) is compact subset of X . If we work in the category TOP A of topological. spaceiwith base point and proper maps presérving base point, wé can give to the set of proper maps f :(S n -M, (X, x 0 ) an equivalence relatioi such that the quotient set . has a group structure . Concretely : P. pointed (n,m)-sphere is a pair . (S,e), where S = Sn -M is
doi:10.5565/publmat_26382_02 fatcat:r56djup7rfej3j5t436nvdm5zu