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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
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