Homological stability of oriented configuration spaces
Let \(F(X,d)\) be the ordered configuration space \(\{(x_1, \dots, x_d)\in X^d: x_i \neq x_j\) if \(i\neq j\}\) and let \(C_d(X)\) denote the quotient space \(F(X,d)/\Sigma_d\), where \(\Sigma_d\) is the symmetric group of \(d\) letters \(\{1,2, \dots,d\}\) and it acts by the permutation of coordinates. The space is called the unordered configuration space of \(d\) distinct points in \(X\) and there are many interesting applications in homotopy theory. For the alternation subgroup \(A_d\subset \Sigma_d\) of index 2, let \(\widetilde C_d(X)\) be the quotient space \(F(X,d)/ A_d\) and it is also called the oriented configuration space of \(d\) distinct points in \(X\). In this paper, the authors study the homological stability of the stabilization map \(\widetilde C_d (X)\to \widetilde C_{d+1} (X)\) for the case \(X=M_g-\)\{finite points\} using several results concerning the homologies of configuration spaces due to Bödigheimer, F. Cohen and L. Taylor. This result may be useful for studying the recent stability phenomena of several moduli spaces.
scientific article; zbMATH DE number 1062467