Divisors and configurations on a surface.

*(English)*Zbl 0686.55012
Algebraic topology, Proc. Int. Conf., Evanston/IL 1988, Contemp. Math. 96, 103-108 (1989).

[For the entire collection see Zbl 0673.00017.]

The space of rational functions on a closed orientable surface, equivalently a suitable space of unordered poles and roots, has been studied by G. Segal [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)]. Related to this are the configuration spaces of unordered k-tuples of points on a space M. Acting by permutation of coordinates, there is a k- plane bundle \[ R^ k\quad \to \quad F(M,k)\times_{\Sigma_ k}\quad R^ k\quad \to \quad F(M,k)/\Sigma_ k, \] called \(\lambda_ k\), where F(M,k) is the space of distinct orderd k-tuples. The order of these bundles, as well as their Thom complexes are of great interest here. This paper determines the order for oriented closed surfaces of genus \(\leq 1\). The authors also analyze the stable homotopy type of the Thom complexes of the multiples of \(\lambda_ k\). The calculations depend on work of these authors, as well as N. Kuhn and L. Taylor, which will appear soon.

The space of rational functions on a closed orientable surface, equivalently a suitable space of unordered poles and roots, has been studied by G. Segal [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)]. Related to this are the configuration spaces of unordered k-tuples of points on a space M. Acting by permutation of coordinates, there is a k- plane bundle \[ R^ k\quad \to \quad F(M,k)\times_{\Sigma_ k}\quad R^ k\quad \to \quad F(M,k)/\Sigma_ k, \] called \(\lambda_ k\), where F(M,k) is the space of distinct orderd k-tuples. The order of these bundles, as well as their Thom complexes are of great interest here. This paper determines the order for oriented closed surfaces of genus \(\leq 1\). The authors also analyze the stable homotopy type of the Thom complexes of the multiples of \(\lambda_ k\). The calculations depend on work of these authors, as well as N. Kuhn and L. Taylor, which will appear soon.

Reviewer: D.W.Kahn