# zbMATH — the first resource for mathematics

On commutativity of connected groups. (English) Zbl 1154.22003
Sb. Math. 197, No. 1, 1-21 (2006); translation from Mat. Sb. 197, No. 1, 3-24 (2006).
The paper contains many conditions of homotopical or shape nature guaranteeing that a given compact connected finite-dimensional topological group $$G$$ is abelian. In particular, this happens if one of the following holds: (1) each homomorphism $$S^3\to G$$ is trivial; (2) the path-connected component of the automorphism group of $$G$$ is a singleton; (3) for every $$g\in G$$ and $$m\geq 2$$ the set $$\{x\in G:x^m=g\}$$ does not contain a topological copy of $$S^2$$ or $$\mathbb RP^2$$; (4) the maps $$\mu_1:(x,y)\mapsto xy$$ and $$\mu_2:(x,y)\mapsto yx$$ are shape equivalent; (5) the inversion map $$(\cdot)^{-1}:G\to G$$ is shape equivalent to a homomorphism; (6) $$\pi_3(G)=0$$; (7) any map $$P\to G$$ from a simply-connected Peano continuum $$P$$ is null homotopic; (8) any homomorphism $$L\to G$$ from a compact simply-connected Lie group $$L$$ is trivial; (9) there is a homomorphism $$G\to A$$ onto an abelian group of dimension $$\dim(A)\geq\dim(G)-2$$.

##### MSC:
 22A05 Structure of general topological groups 22B05 General properties and structure of LCA groups 22C05 Compact groups 54C56 Shape theory in general topology
##### Keywords:
compact topological group; abelian group
Full Text: