Cohomotopy of Lie groups (Q1191030)

The purpose of this paper is to study the set \(\text{Cdg}(X,n)=\text{Cdg}([X,S^ n])\) when \(X=G\) is a compact simply connected simple Lie group, where \(\text{Cdg}: [X,S^ n]\to\text{Hom}(\pi_ n(X),\pi_ n(S^ n))\) assigns the induced homotopy homomorphism \(f_ *\) to the homotopy class of a map \(f: X\to S^ n\). To estimate \(\text{Cdg}(X,n)\) the authors introduce an invariant \(\text{cdg}(X,n)\) and its stable version \(^ s\text{cdg}(X,n)\), which are non-negative integers or infinity, such that \(^ s\text{cdg}(G,3)\) coincides with another invariant, which was denoted by the first author by \(\text{cd}(G)\) [Osaka J. Math. 26, 759-773 (1989; Zbl 0705.55005)]. The authors denote by \(\text{cdg}_ p(X,n)\) the exponent of a prime number \(p\) in the prime power decomposition of \(\text{cdg}(X,n)\) when \(0<\text{cdg}(X,n)<\infty\); \(\text{cdg}_ p(X,n)=0\) when \(\text{cdg}(X,n)=0\). Similarly \(^ s\text{cdg}_ p(X,n)\) is defined. The following theorems are proved: Theorem 1. If \(G\) is a compact simply connected Lie group such that \(G=G_ 1\times\dots\times G_ t\) with \(G_ i\) a compact simply connected Lie group, then \(\text{cdg}(G,n)\) and \(^ s\text{cdg}(G,n)\) are finite and the following statements are equivalent for any prime number \(p\). (1) \(\text{cdg}_ p(G,3)=0\), (2) \(^ s\text{cdg}_ p(G,3)=0\), (3) \(\text{cdg}_ p(G_ i,3)=0\) for all \(i\), (4) \(^ s\text{cdg}_ p(G_ i,3)=0\) for all \(i\), (5) \(G_ i\) is \(p\)-regular for every \(i\), (6) \(G\) is \(p\)-regular, (7) \(\text{cdg}_ p(G,n)=0\) for all \(n\). Theorem 2. If \(G\) is a compact simply connected simple Lie group, then \(\text{Cdg}(G,n)\) is a subgroup of \(\text{Hom}(\pi_ n(G),\pi_ n(S^ n))\) of maximal rank. Indeed \(\text{Cdg}(G,n)\) is \(\text{cdg}(G,n)\mathbb{Z}\{s_ 1'\}\oplus c\mathbb{Z}\{s_ 2'\}\) if \((G,n)=(\text{Spin}(4m),4m-1)\) and \(\text{cdg}(G,n)\cdot\text{Hom}(\pi_ n(G),\pi_ n(S^ n))\) otherwise. Here \(\pi_{4m- 1}(\text{Spin}(4m))=\mathbb{Z}\{s_ 1\}\oplus\mathbb{Z}\{s_ 2\}\) and \(s_ i'\) is the dual element to \(s_ i\); \(c\) is 1 if \(m\leq 2\) and 2 if \(m\geq 3\); \(\text{cdg}(G,n)\) is non-zero if and only if \(n\in\{n_ 1,\dots,n_ r\}\), where \(H^*(G;Q)\cong H^*(\prod^ r_{i=1}S^{n_ i};Q)\). All spaces are path-connected with base point and all maps preserve base points. The base point of any \(H\)-space is the unit of it.