Holomorphic spheres in loop groups and Bott periodicity (Q1840697)

The authors investigate a holomorphic version of the Bott periodicity theorem: there is a natural homotopy equivalence \(\mathbb{Z}\times BU\to\Omega U\), where \(BU= \varinjlim BU(n)\) is the limit of the classifying spaces and \(\Omega U= C^\infty(S^1, U)\) is the loop space of the infinite unitary group \(U= \varinjlim U(n)\). For that purpose the authors consider the space \(\text{Hol}_k(P^1, \text{Gr}_m(\mathbb{C}^n))\) of holomorphic degree \(k\)-maps of the Riemann sphere \(P^1\) into the infinite-dimensional Grassmannian \(\text{Gr}_m(\mathbb{C}^n)\) of \(m\)-dimensional subspaces of \(\mathbb{C}^n\) and define \(\text{Hol}_k(P^1, BU)\) as the \(\varinjlim_{m,n} \text{Hol}_k(P^1, \text{Gr}_m(\mathbb{C}^n))\) with a natural topology. The authors prove that for each positive integer \(k\) there is a natural homotopy equivalence \(\text{Hol}_k(P^1, BU)\to BU(k)\). A similar result is obtained for \(\text{Hol}_k(P^1, BU(n))\) for each \(n\).