Symmetric Joins and Weighted Barycenters

Abstract

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 that 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.

Homotopy groups of diagonal complements

“…We have already indicated that Q n,k ≃ B n−1 (R k , n). On the other hand, according to Theorems 1.1, 1.3 and 1.5 of [16], we have that…”

“…The case k = 1 is trivial. We let k ≥ 2 and invoke some main results from [16,17]. Let S be the unit sphere as in Lemma 2.2 and let Q n,k be its quotient under the S n -action.…”

Min–max schemes for SU(3) Toda systems

“…Assuming by contradiction that α ρ ≤ − L 2 , there would exist a map π ∈ Π Φ with supσ ∈Km II(π(σ)) ≤ − 3 8 L. Then, since Proposition 2.7 applies with L 4 , writingσ = (σ, t), with σ ∈ Σ m , the map t → Ψ m • π(·, t) would be an homotopy in Σ m between Ψ m • Φ and a constant map. But this is impossible since Σ m is non-contractible (see [27]) and since Ψ m • Φ is homotopic to the identity on Σ m , by Proposition 2.8. Therefore, we deduce α ρ > − L 2 .…”

“…Remark 3.6 Since each γ i is topologically a circle, it follows from Proposition 3.2 in [10] that (γ 1 ) k (S 1 ) k is homeomorphic to S 2k−1 and (γ 2 ) l to S 2l−1 (in [27] it was proved previously a homotopy equivalence). As it is well-known, the topological join S m * S n is homeomorphic to S m+n+1 (see for example [23]), and therefore (γ 1 ) k * (γ 2 ) l is homeomorphic to the sphere S 2k+2l−1 .…”

“…The above set of measures, which is naturally endowed with the weak topology of distributions, does not have a smooth structure for k ≥ 2 (while Σ 1 is homeomorphic to Σ): it is indeed a stratified set, namely union of open manifolds of different dimensions (see [27] for further characterizations, especially for what concerns the Betti numbers). The basic property used in [21] and [20] is that Σ k is non-contractible, which allows to define proper min-max schemes to attack the existence problem.…”