## 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.

### MSC:

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

### Citations:

Zbl 0292.55018; Zbl 0286.55015; Zbl 0638.55015; Zbl 0654.55014
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