an:00009951
Zbl 0741.55005
Cohen, F. R.; Cohen, R. L.; Mann, B. M.; Milgram, R. J.
The topology of rational functions and divisors of surfaces
EN
Acta Math. 166, No. 3-4, 163-221 (1991).
00221282
1991
j
55P42 55P99
holomorphic maps; complex manifolds; homotopy types; spaces of divisors; function space; stable homotopy types; moduli spaces of monopoles
From the authors' introduction: ``Spaces of holomorphic maps between complex manifolds have played a fundamental role in such diverse branches of mathematics as analysis, differential geometry, topology, mathematical physics, and linear control theory. In seminal work, \textit{G. Segal} [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)] studied the homotopy types of the spaces of holomorphic functions of the 2-sphere \(S^ 2\), of closed surfaces of higher genus, and of spaces of divisors of these surfaces. In particular he showed that the space of holomorphic functions of degree \(k\) fills out the homotopy type of an appropriate function space in a stable range of dimensions (roughly up to dimension \(k-2g\), where \(g\) is the genus). In this paper we continue Segal's program by describing the entire stable homotopy types of these spaces in terms of the homotopy types of more familiar spaces.''
The main result of the paper concerns the homotopy type of the space \(Rat_ k\), the space of based holomorphic self-maps of the Riemann sphere having degree \(k\). Segal determined the homotopy type of \(Rat_ k\) through dimension \(k\) [loc. cit.]. This paper shows that \(Rat_ k\) has the stable homotopy type of the Eilenberg-MacLane space \(K(\beta_ 2k,1)\), where \(\beta_ n\) is Artin's braid group on \(n\)-strings. Further, this stable type is explicitly described in terms of well-known constructions. Connections with spaces of divisors and with moduli spaces of monopoles are also given.
P.J.Kahn (Ithaca)
Zbl 0427.55006