When certain natural maps are equivalences

Richard Holzsager
1972 Pacific Journal of Mathematics  
This paper classifies those spaces for which a certain natural map is a homotopy equivalence. Five cases are considered: X-> SP°°X 9 the map from a space to its infinite symmetric product; Ω^S^X-tSP^X, the map from the "infinite loop space of the infinite suspension" to the infinite symmetric product; X-> Ω n S n X 9 the map from a space to the w-fold loop space of the n-ίold suspension; S n Ω n X->X, the map from the n ίolά suspension of the %-fold loop space of a space to the space itself;
more » ... Ω^S^X, the map from a space to the infinite loop space of the infinite suspension. Under the assumption (made throughout) that the spaces have the homotopy type of connected CW-complexes, these are actually questions about relationships among the homotopy groups, stable homotopy groups and homology groups. The proofs are mostly algebraic.
doi:10.2140/pjm.1972.42.69 fatcat:pcbpk6fk2jhpvp7lhtoyprpjvy