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The topology of the space of rational maps into generalized flag manifolds. (English) Zbl 0844.57037
This paper deals with a complex semi-simple Lie group \(G\) and parabolic subgroup \(P\). The space \(\text{Rat} (G/P)\) of holomorphic maps from \(\mathbb{P}^1\) to \(G/P\) has components \(\text{Rat}_{\mathbf k} (G/P)\) labelled by a multidegree \({\mathbf k} = (k_1,\dots, k_n) \in \pi_2(G/P)\), whose \(j\)th component \(k_j\) is the intersection number of the image of \(S^2\) with the closure of a codimension-1 Bruhat cell. The components of \(\Omega^2(G/P)\) can be labelled similarly, and there is an inclusion \(\text{Rat}_{\mathbf k}(G/P) \to \Omega^2_{\mathbf k} (G/P)\). The main theorem states that this map is a homotopy equivalence through dimension \(cl({\mathbf k}) - 1\) or \(cl({\mathbf k}) - 2\), where \(l({\mathbf k}) = \min(k_i)\) and \(c \leq {1\over 2}\) is a positive constant which depends only on \(G/P\). The proof involves analyzing homology groups and fundamental groups. The constant \(c\) depends on codimensions in a stratification of \(\text{Rat}_{\mathbf k} (G/P)\).

MSC:
57R99 Differential topology
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