Finite words in a based space, with occurrences of its basepoint deleted. Word concatenation defines a multiplication. For connected based CW spaces, the natural map to is a weak homotopy equivalence. With free integral homology, the Bott–Samelson theorem identifies its homology algebra with the tensor algebra on reduced homology.
For a connected based CW complex with free integral homology, the Pontryagin ring of its looped reduced suspension is the indicated tensor algebra. The generator inclusion comes from , and multiplication corresponds to concatenation of words in the James reduced product.
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James reduced product is a construction in algebraic topology, specifically in the context of homotopy theory. It is named after the mathematician I. M. James, who introduced it in his work on fiber spaces and homotopy groups. The James reduced product addresses the issue of a certain type of product in the category of pointed spaces (spaces with a distinguished base point), particularly when working with spheres. The concept is useful when studying the stable homotopy groups of spheres.