Bott maps and the complex projective plane: a construction of R. Wood’s equivalences

Abstract

Let U(oo), O(oo) and Sp(oo) be the direct limits of the finite-dimensional unitary, orthogonal and symplectic groups under inclusion, and let P 2 C be the complex projective plane. Then, by a result of R. Wood in X-theory, there exist homotopy equivalences from U(oo) to the space of based maps P 2 C -> O(oo), and to the space of based maps P 2 C -> Sp(oo). In this paper we give an explicit construction of such homotopy equivalences, and prove Wood's theorem by using classical results of R. Bott and elementary homotopy theory.