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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.
Reviewer: D.W.Kahn

##### MSC:
 55R50 Stable classes of vector space bundles in algebraic topology and relations to $$K$$-theory 54C35 Function spaces in general topology