Homotopy classification of self-maps of \(BG\) via \(G\)-actions. I. (English) Zbl 0758.55004

The authors study here the homotopy type problem for \(\text{map}(BG,BG')\), where \(G\) and \(G'\) are compact connected simple Lie groups. Their main result is the isomorphism of monoids with zero \[ \text{Rep}(G,G)\times\{k\geq 0: k=0 \text{ or }(k,| W|)=1\}\buildrel\cong\over\longrightarrow [BG,BG], \] which sends \((\alpha,k)\) to \(\psi^ k\circ B\alpha\). Here \(\text{Rep}(G,G')=\text{Hom}(G,G')/\text{Inn} G'\), \(W\) is the Weyl group, \(M_ 1\wedge M_ 2=M_ 1\times M_ 2/\langle (x_ 1,0)\sim (0,0)\sim(0,x_ 2): x_ i\in M_ i\rangle\) and \(\psi^ k\) is the unique unstable Adams operation of degree \(k\) (i.e. the unique class satisfying \(H^{2i}(\psi^ k: {\mathbf Q})=k^ i\) for each \(i\geq 0\)). The above map was shown to be onto by J. R. Hubbuck [Q. J. Math., Oxf., II. Ser. 25, 113-133 (1974; Zbl 0292.55018); New developments in Topology, Proc. Symp. Alg. Topol., Oxford 1972, 33-41 (1974; Zbl 0286.55015)] and Mahmud; and K. Ishiguro [Math. Proc. Camb. Philos. Soc. 102, 71-75 (1987; Zbl 0638.55015)] showed that unstable Adams operations of type \(\psi^ k\) exist only for \(k=0\) or \((k,| W|)=1\). The authors also prove that there are maps \(e_ 0: BG\to \text{map}(BG,BG)_ 0\) and \(e_ f: BZ(G)\to \text{map}(BG,BG)_ f\) \((f\nsim 0)\) which induce homomorphisms of homology with arbitrary finite coefficients: this result was obtained for \(G=SU(2)\) by W. G. Dwyer and G. Mislin [Algebraic topology, Proc. Symp., Barcelona 1986, Lect. Notes Math. 1298, 82-89 (1987; Zbl 0654.55014)].
The above classification result is proved using a new homotopy decomposition of \(BG\) obtained by constructing certain finite-dimensional acyclic \(G\)-complexes.


55P15 Classification of homotopy type
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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