Order of the canonical vector bundle over configuration spaces of spheres
For a vector bundle \(\xi\) let \(\xi^{\oplus t}\) denote the \(t\)-fold Whitney sum with itself and let \(\epsilon^s\) denote the trivial vector bundle of dimension \(s\). If there exists a positive integer \(t\) such that \(\xi^{\oplus t}\) is trivial, \(\xi\) is said to have a finite order and the smallest such \(t\) is called the order of \(\xi\). Similarly, \(\xi\) is said to be stably trivial if there exist positive integers \(t\) and \(s\) such that \(\xi\oplus \epsilon^t\) is isomorphic to \(\epsilon^s\). If there exists a positive integer \(t\) such that \(\xi^{\oplus t}\) is stably trivial then \(\xi\) is said to have finite stable order and the smallest such \(t\) is called the stable order of \(\xi\). For a relative CW complex \((M,M_0)\) and a positive integer \(k\geq 2\), let \(F(M| M_0,k)\) denote the configuration space defined by \[ F(M| M_0,k)= \{(x_1,\dots ,x_k)\in M^k:x_i\not= x_j \text{ if }i\not= j \text{ and } x_i\in M_0 \text{ for some }i\} \] The symmetric group \(\Sigma_k\) of \(k\) letters acts on \(F(M| M_0,k)\) freely by the usual permutation of coordinates, and let \(F(M| M_0,k)/\Sigma_k\) denote its orbit space. Let \(\xi_{M| M_0,k}\) denote the \(k\)-dimensional real vector bundle given by \(F(M|M_0,k)\times_{\Sigma_k}\mathbb{R}^k\to F(M|M_0,k)/\Sigma_k\) and we write \(\xi_{M,k}\) when \(M=M_0\). For a positive integer \(t\) let \(\rho (t)\) denote the number of positive integers \(1\leq k\leq t\) such that \(k\equiv 0,1,2,4\) mod \(8\), and let \(a_{m,k}\) denote the positive integer defined by \(a_{m,k}=2^{\rho (m-1)}\prod_{3\leq p\leq k,p\text{ prime }}p^{\lfloor \frac{m-1}{2}\rfloor}.\) In this paper, the author studies the order and stable order of \(\xi_{M| M_0,k}\) when \(M=\coprod_nS^m\) and \(M_0\) is a non-empty CW-subcomplex of \(M\). More precisely, he generalizes the result obtained by \textit{F. R. Cohen} et al. [Pac. J. Math. 104, 47--54 (1983; Zbl 0513.55007)] and he proves that the order and the stable order of \(\xi_{\coprod_nS^m|M_0,k}\) are equal and that the order of \(\xi_{\coprod_nS^m,k}\) is either \(a_{m,k}\) or \(2^{\rho (m)-\rho (m-1)}a_{m,k}\). Moreover, if \(k\) is not prime, he proves that the order of \(\xi_{\coprod_nS^m|M_0,k}\) is either \(a_{m,k}\) or \(2^{\rho (m)-\rho (m-1)}a_{m,k}\). In particular, when either \((n,m)=(1,2)\) and \(k\geq 2\) is even or \(n\geq 2\), he proves that the order of \(\xi_{\coprod_nS^m,k}\) is \(2^{\rho (m)-\rho (m-1)}a_{m,k}\) and that the order of \(\xi_{\coprod_nS^m|M_0,k}\) is \(2^{\rho (m)-\rho (m-1)}a_{m,k}\) if \(k\) is not prime.
scientific article; zbMATH DE number 6947023