Holomorphic spheres in loop groups and Bott periodicity. (English) Zbl 1064.58010

Cohen, Ralph L.; Lupercio, Ernesto; Segal, Graeme B.

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

Keywords:

spaces of holomorphic maps; Riemann sphere; infinite-dimensional Grassmann manifolds