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The topology of rational maps to Grassmannians and a homotopy theoretic proof of the Kirwan stability theorem. (English) Zbl 0795.55008
Tangora, Martin C. (ed.), Algebraic topology, Oaxtepec 1991. Proceedings of an international conference on algebraic topology, held July 4-11, 1991 in Oaxtepec, Mexico. Providence, RI: American Mathematical Society. Contemp. Math. 146, 251-275 (1993).
Summary: Let \(\text{Rat}_ k (\mathbb{G}_{n,m})\) be the space of based holomorphic maps of degree \(-\kappa\) from the Riemann sphere to the complex Grassmannian manifold of \(n\) planes through the origin in \(\mathbb{C}^{n+m}\). In [J. Differ. Geom. 33, No. 2, 301-324 (1991; Zbl 0736.54008)] we studied the geometry and computed the homology of \(\text{Rat}_ k (\mathbb{G}_{n,m})\). This homology computation was quite indirect and depended in part on a stability result of F. C. Kirwan [Ark. Math. 24, 221-275 (1986; Zbl 0625.14026)] which used techniques and results from algebraic geometry. In this note we give an independent, self-contained proof of the main topological results of the authors [loc. cit.], that uses only standard results in homotopy theory and along the way give a more precise description of \(H_ *(\text{Rat}_ k (\mathbb{G}_{n,m}))\) than was obtained in [the authors, loc.cit.]. As a corollary we obtain a much sharper version of Kirwan’s theorem.
For the entire collection see [Zbl 0780.00041].

MSC:
55P35 Loop spaces
49J15 Existence theories for optimal control problems involving ordinary differential equations
81T13 Yang-Mills and other gauge theories in quantum field theory
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