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Holomorphic spheres in loop groups and Bott periodicity. (English) Zbl 1064.58010
Yau, S.T. (ed.), Surveys in differential geometry. Papers dedicated to Atiyah, Bott, Hirzebruch and Singer . Somerville, MA: International Press (ISBN 1-57146-069-1/hbk). Surv. Differ. Geom., Suppl. J. Differ. Geom. 7, 83-106 (2000).
Summary: We study the topology of spaces of holomorphic maps from the Riemann sphere $$\mathbb{P}^1$$ to infinite-dimensional Grassmann manifolds and to loop groups. Included in this study is a complete identification of the homotopy types of $$\text{Hol}_k(\mathbb{P}^1,BU(n))$$ and of $$\text{Hol}_k(\mathbb{P}^1,\Omega U)$$, where the subscript $$k$$ denotes the degree of the map. These spaces are shown to be homotopy equivalent to the $$k$$th Mitchell-Segal algebraic filtration of the loop group $$\Omega U(n)$$, and to $$BU(k)$$, respectively.
For the entire collection see [Zbl 1044.53002].

##### MSC:
 58D15 Manifolds of mappings