Presentations of the First Homotopy Groups of the Unitary Groups

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n ≥ 2 we supply n homotopic maps that each represent the (n − 1)!-th power of a suitable generator of π2nSU(n) ≈ Zn!. The product of these n commuting maps is the constant map to the identity matrix. Introduction The homotopy groups of compact Lie groups have been of continuous interest since the discovery of homotopy groups at around 1935. There is now a tremendous amount of computational tools available and many groups have been determined. On the other hand, intellectually and practically satisfying presentations of these groups are only known in comparatively few cases. Our goal in this paper is to describe such presentations for the first homotopy groups of the unitary groups. For the stable groups we mainly, but not entirely, review some known results and procedures in an easily accessible and most explicit way. Particular emphasis is given to the last stable groups π2n−1U(n). A highlight of this part is a strikingly simple formula for a minimal embedding of S into SU(3) that represents a generator of π5SU(3) and has a natural interpretation in terms of the complex cross product. The main achievement of the paper concerns the first nonstable homotopy groups π2nSU(n) ≈ Zn!. These groups played an important role in the first proofs of the fact that the only parallelizable spheres are S, S, and S. We figure in an elementary and explicit way how a suitable generator of π2nSU(n) becomes nullhomotopic in the n!-th power. Namely, we supply n homotopic maps that each represent the (n−1)!-th power of the generator. The product of these n commuting maps is the constant map to the identity matrix. The generators of π2nSU(n) are then used to produce similar presentations of the homotopy groups π2nSU(n − 1) with even n. Finally, we answer the simple question whether a map S → U(n) is homotopic to its transposed or its complex conjugate for r ≤ 2n+2 with the help of the explicit maps described before. As applications we obtain presentations of certain stable homotopy groups of the symplectic groups and a structure theorem for certain nonstable homotopy groups of the symmetric spaces SU(n)/SO(n). 2000 Mathematics Subject Classification. Primary 57T20. The joint work of the authors was supported by CNPq and the International Bureau of the BMBF in the scope of the former CNPq/GMD-agreement. 1 2 THOMAS PÜTTMANN AND A. RIGAS Throughout this paper we use the well-known fact that the homotopy group πk(G) of a compact connected Lie group G is isomorphic to the group of free homotopy classes of maps S → G. Here, the product between two free homotopy classes is given by multiplying the representing maps value by value with the product of G. We also often use the elementary fact that the inclusions SU(n) → U(n) induce isomorphisms between πrSU(n) and πrU(n) for r ≥ 2. 1. The stable homotopy groups of the unitary groups 1.1. Bott periodicity. It has been known since around 1940 that the inclusion of U(n) into U(n+ 1) induces an isomorphism between the homotopy groups πrU(n) and πrU(n+1) if r < 2n. The homotopy groups in this range are called stable. Their simple structure became visible at the end of the 50’s by Bott’s famous periodicity theorem [3]: The stable groups πrU are trivial if r is even and isomorphic to Z if r is odd. In fact, Bott constructed isomorphisms πrU(n) → πr+2SU(2n) for r < 2n and thus all stable groups are determined by π1U(1) ≈ Z and the trivial group π2U(2). The periodicity isomorphisms can be given in the following explicit way: One assigns to a map θ : S → U(n) the map B(θ) : S → SU(2n) defined on the unit sphere in C × R by B(θ) ( w x )