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This paper classifies those
spaces for which a certain natural map is a homotopy equivalence. Five cases are
considered:
X → SP∞X, the map from a space to its infinite symmetric product;
Ω∞S∞X → SP∞X, the map from the “infinite loop space of the infinite
suspension” to the infinite symmetric product;
X → ΩnSnX, the map from a space to the n-fold loop space of the n-fold
suspension;
SnΩnX → X, the map from the n-fold suspension of the n-fold loop space of a
space to the space itself;
X → 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.
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