Symmetric Joins and Weighted Barycenters

By S. Kallel, Rym Karoui. 2006

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Abstract Given a space X, we study the homotopy type of Bn(X) the space obtained as the “union of all (n−1)-simplexes spanned by points in X” or the space of “formal barycenters of weight n or less” of X. This is a space encountered in non-linear analysis under the name of space of barycenters or in differential geometry in the case n = 2 as the space of chords. We first relate this space to a more familiar symmetric join construction and then determine its stable homotopy type in terms of the symmetric products on the suspension of X. This leads to a complete understanding of the homology of Bn(X) as a functor of X, and to an expression for its Euler characteristic given in terms of X. A sharp connectivity theorem is also established. Finally the case of spheres S is studied in details and the homotopy type of Bn(S ) is described, generalizing in this way an early and beautiful result of James, Thomas, Toda and Whitehead.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3e863498a5c09748b2025c86fcffd478a06b8ba7