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

Minato Yasuo

Vol. 128 (1987), No. 2, 379–390

DOI: 10.2140/pjm.1987.128.379

Abstract

Let U(∞), O(∞) and Sp(∞) be the direct limits of the finite-dimensional unitary, orthogonal and symplectic groups under inclusion, and let P₂C be the complex projective plane. Then, by a result of R. Wood in K-theory, there exist homotopy equivalences from U(∞) to the space of based maps P₂C → O(∞), and to the space of based maps P₂C → Sp(∞). 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.

Mathematical Subject Classification 2000

Primary: 55P10

Milestones

Received: 17 March 1986

Published: 1 June 1987