Vol. 42, No. 1, 1972

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When certain natural maps are equivalences

Richard Allan Holzsager

Vol. 42 (1972), No. 1, 69–75

This paper classifies those spaces for which a certain natural map is a homotopy equivalence. Five cases are considered:

X SPX, the map from a space to its infinite symmetric product;

ΩSX SPX, 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 ΩSX, 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.

Mathematical Subject Classification
Primary: 55D15
Received: 24 May 1971
Published: 1 July 1972
Richard Allan Holzsager