Codegree of simple Lie groups. (English) Zbl 0705.55005

Ōshima, Hideaki

Osaka J. Math. 26, No. 4, 759-773 (1989).

Given a based map f: \(S^ n\to X\), the generalized reduced cohomology theory E provides us with the notion of the codegree \(cd(f;E^ m)\) of f, defined to be the cardinal number of the cokernel of the composite \[ E^{m+n}(X)\to^{f^*}E^{m+n}(S^ n)\twoheadrightarrow E^ m(S^ 0)/Tor. \] The author’s main concern is \(cd(f)=cd(f;\pi^ 0_ s)\) where \(\pi^ 0_ s\) is the stable cohomotopy functor, especially the codegree \(cd(G)=cd(f)\) of a map f generating \(\pi_ 3(G)\cong {\mathbb{Z}}\) for a simply connected simple Lie group G. In this connection his main results are as follows. 1) \(cd_ p(G)\), the exponent of the prime p in cd(G), is 0 if and only if G is p-regular. 2) \(cd_ p(G)=1\) if G is not p-regular but quasi p-regular. 3) \(cd_ p(SU(n))=\max \{i;p^ i<n\}\) if \(p>2\) and \(n\geq 3\); \(cd_ p(SO(2n+1)=cd_ p(Spin(2n+1))=cd_ p(Sp(n))\) for \(p>2\) and \(n\geq 1\); \(cd_ p(SO(2n))=cd_ p(Spin(2n))=cd_ p(Spin(2n-1))\) for \(p>2\) and \(n\geq 3\); \(cd(Sp(n))=cd(SU(2n))\). 4) \(cd_ 7(E_ 8)=cd_ 7(E_ 7)=cd_ 5(E_ 7)=1\); \(cd(G_ 2)=cd(Spin(7))=cd(Spin(8))=cd(SO(7))=2\cdot cd(SU(7))=2^ 3\cdot 3\cdot 5\) and so on. These are deduced with the help of the work of M. C. Crabb and K. Knapp [Math. Ann. 282, 395-422 (1988; Zbl 0627.57026)] by using the fact that the codegree coincides with that of a canonical bundle over a projective space. Some of the codegrees for small p are left undetermined.

Reviewer: Y.Nomura

MSC:

55Q55	Cohomotopy groups
57T10	Homology and cohomology of Lie groups
55N15	Topological \(K\)-theory

Keywords:

cohomology theory; codegree; stable cohomotopy; Lie group; exponent; p- regular; canonical bundle; projective space

Citations:

Zbl 0649.57037; Zbl 0627.57026