ASIAN J. MATH. c  1999 International Press Vol. 3, No. 4, pp. 801–818, December 1999 005 HOLOMORPHIC SPHERES IN LOOP GROUPS AND BOTT PERIODICITY ∗ RALPH L. COHEN † , ERNESTO LUPERCIO ‡ , AND GRAEME B. SEGAL § Abstract. In this paper we study the topology of spaces of holomorphic maps from the Riemann sphere P 1 to infinite dimensional Grassmanian manifolds and to loop groups. Included in this study is a complete identification of the homotopy types of Hol k (P 1 ,BU (n)) and of Hol k (P 1 , ΩU ), 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 ΩU (n) [7], and to BU (k), respectively. Introduction. One of the most important theorems in Topology and Geometry is the “Bott Periodicity Theorem”. In its most basic form it states that there is a natural homotopy equivalence, β : Z × BU ≃ −−−−→ ΩU. Here U is the infinite unitary group, U = lim −→n U (n), BU = lim −→n BU (n) is the limit of the classifying spaces, and ΩU = C ∞ (S 1 ,U ) is the space of smooth, basepoint preserving loops. Here and throughout the rest of this paper all spaces will assumed to be equipped with a basepoint, and all maps and mapping spaces will be basepoint preserving. If we input the fact that U ≃ ΩBU , Bott periodicitiy states that there is a natural homotopy equivalence β : Z × BU ≃ −−−−→ Ω 2 BU = C ∞ (S 2 ,BU ). In a paper which first pointed to the deep relationship between the index theory of Fredholm operators and Algebraic Topology, Atiyah [1], defined a homotopy inverse to the Bott map β, which can be viewed as a map ¯ ∂ : C ∞ (S 2 ,BU ) ≃ −−−−→ Z × BU. This map was defined by studying the index of the family of operators obtained by coupling the ¯ ∂ operator to a smooth map from S 2 to a Grassmannian. Since the mapping spaces C ∞ (S 2 ,BU ) and ΩU both have path components nat- urally identified with the integers, we denote by C ∞ (S 2 ,BU ) k the path components consisting of degree k - maps. Thus Bott periodicity, together with Atiyah’s results says that for each integer k, there is a natural homotopy equivalence, ¯ ∂ : C ∞ (S 2 ,BU ) k ≃ −−−−→ BU . The goal of this paper is to prove a holomorphic version of this result. We first note however that the homotopy type of BU has many different models, several of which carry a holomorphic structure. For the purposes of this paper we think of BU * Received July 30, 1999, accepted for publication January 10, 2000. † Dept. of Mathematics, Stanford University, Stanford, California 94305, USA (ralph@math. stanford.edu). The research of this author was partially supported by a grant from the NSF. ‡ Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA (lupercio @math.lsa.umich.edu). The research of this author received partial support from CONACYT. § Dept. of Pure Mathematics and Mathematical Statistics, Cambridge University, Cambridge, England (G.B.Segal@dpmms.cam.ac.uk). 801