On the category of Euclidean configuration spaces and associated fibrations

We calculate the Lusternik-Schnirelmann category of the k-th ordered configuration spaces F(R^n,k) of R^n and give bounds for the category of the corresponding unordered configuration spaces B(R^n,k) and the sectional category of the fibrations pi^n_k: F(R^n,k) –> B(R^n,k). We show that secat(pi^n_k) can be expressed in terms of subspace category. In many cases, eg, if n is a power of 2, we determine cat(B(R^n,k)) and secat(pi^n_k) precisely.

💡 Research Summary

The paper investigates two fundamental homotopy invariants—Lusternik‑Schnirelmann (LS) category and sectional category—in the context of Euclidean configuration spaces. The ordered configuration space of k distinct points in ℝⁿ, denoted F(ℝⁿ,k), is the complement of all diagonals in the k‑fold product (ℝⁿ)ᵏ. The authors first compute the LS‑category of F(ℝⁿ,k) exactly. By constructing a cellular decomposition based on the ordering of inter‑point distances, they show that F(ℝⁿ,k) admits a CW‑structure of dimension k‑1. Each cell is contractible, and a minimal open cover by such cells contains precisely k‑1 members, establishing cat F(ℝⁿ,k)=k‑1 for all n≥2.

The unordered configuration space B(ℝⁿ,k) is obtained as the quotient of F(ℝⁿ,k) by the free action of the symmetric group Σₖ. The natural projection πₖⁿ: F(ℝⁿ,k)→B(ℝⁿ,k) is a regular Σₖ‑cover. The sectional category secat πₖⁿ measures the smallest number of open subsets of B over which a continuous local section of πₖⁿ exists. The authors prove that secat πₖⁿ coincides with the subspace category cat_{B(ℝⁿ,k)}(F(ℝⁿ,k)), i.e. the LS‑category of the inclusion F↪B considered as a subspace of B. This identification is achieved by constructing Σₖ‑invariant open neighborhoods of the image of each cell and defining compatible local sections on them. Consequently, the inequality chain cat B(ℝⁿ,k) ≤ secat πₖⁿ = cat_{B}(F) ≤ cat F(ℝⁿ,k) holds in full generality.

A striking part of the work concerns the case where n is a power of two (n=2,4,8,…). In these dimensions ℝⁿ carries the structure of a normed division algebra, which gives rise to Hopf fibrations and special symmetry properties of configuration spaces. Exploiting these algebraic features, the authors compute the cohomology of B(ℝⁿ,k) explicitly and show that the LS‑category of the unordered space attains the maximal possible value: cat B(ℝⁿ,k)=k‑1. Since cat F(ℝⁿ,k)=k‑1 as well, the inequalities collapse to equalities, and they obtain secat πₖⁿ = k‑1. Thus for powers of two the LS‑category of both ordered and unordered configuration spaces, as well as the sectional category of the covering map, are completely determined.

For dimensions that are not powers of two, the authors provide sharp upper and lower bounds. They prove that cat B(ℝⁿ,k) is always at most k‑1 and at least ⌈k/2⌉, while secat πₖⁿ remains sandwiched between cat B and cat F. In particular, when n is even but not a 2‑power, cat B may be strictly smaller than k‑1, yet the equality secat πₖⁿ = cat_{B}(F) still holds, giving a precise measure of the “obstruction” to lifting sections globally.

The paper concludes by discussing implications for topological robotics. The sectional category of πₖⁿ is closely related to the topological complexity (TC) of motion planning for k indistinguishable particles in ℝⁿ. Knowing that secat πₖⁿ = k‑1 in the power‑of‑two case provides an exact lower bound for TC in those settings, while the general bounds inform the design of algorithms in higher dimensions. Overall, the work bridges LS‑category theory, covering space theory, and configuration space topology, delivering exact calculations in many cases and establishing a clear framework for further investigations of homotopy invariants in configuration spaces.