Spaces and equations. (English) Zbl 0968.08004

The author proves that, for many topological spaces \(A\), \(A\) carries the structure of a topological \(\Sigma\)-algebra iff an algebraic theory \(\Sigma\) is undemanding. This condition means that \(\Sigma\) is trivial in the sense that it is satisfied on every set by interpreting operation symbols either as projection functions or as constant functions. The proof uses either the fundamental group of \(A\) or the cohomology ring of \(A\). Since these constructions are homotopy-invariant, the author’s result remains valid if one takes satisfaction of equations up to homotopy. The paper thus offers a far-reaching generalization of a celebrated theorem of Adams asserting that the only spheres that are \(H\)-spaces are \(S^1\), \(S^3\) and \(S^7\).


08B05 Equational logic, Mal’tsev conditions
22A30 Other topological algebraic systems and their representations
18C05 Equational categories
57T25 Homology and cohomology of \(H\)-spaces
57T99 Homology and homotopy of topological groups and related structures
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