The space of loops on configuration spaces and the Majer-Terracini index
The authors introduce the concept of \(\Gamma\)-category and use it to obtain a variant of a theorem of Majer and Terracini showing that some functional \(f\) has an unbounded sequence of critical values. More precisely, let \(\Gamma= \{\Gamma_\gamma\}\) be a finite open cover of \(Y\) and \(A\) be a subset of \(Y\) such that \(A\subset \Gamma_\gamma\) for every \(\gamma\). Then \(\Gamma\text{-cat }A\leq p\) if \(A\) may be expressed as a union \(A=\bigcup_\gamma A_\gamma\) with \(\text{cat}_{\Gamma_\gamma}A_\gamma\leq p\) for every \(\gamma\) where \(\text{cat}_{\Gamma_\gamma} A_\gamma\) is the usual Lyusternik-Shnirel'man category for the inclusion \(A_\gamma\subset \Gamma_\gamma\). Let \(X\) be a given ANR and \(F_n(X)\) the \(n\)th configuration space of \(X\). The authors define the following open subsets in \(X^n\) \(F_n(i,j)= \{x\in X^n\mid x_i\neq x_j\) for a given \(i\neq j\}\). Then \(\bigcap_{i<j} F_n(i,j)= F_n(X)\). Applying the free loop functor, they obtain a collection of open sets \(\Lambda\Gamma= \{\Lambda F_n(i,j)\}\), and they show that \(\Lambda F_n (\mathbb{R}^k)\), \(n\geq 2\), contains compact subsets of arbitrary high \(\Lambda\Gamma\)-category. Section 2 of the paper presents the general theory of \(\Omega F_n(\mathbb{R}^k)\) and \(\Lambda F_n(\mathbb{R}^k)\) and gives the Hopf algebra structure of \(H(\Lambda F_n(\mathbb{R}^k))\) in terms of generators and relations.
scientific article; zbMATH DE number 1270100
scientific article