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Non-G-equivalent Moore G-spaces of the same type. (English) Zbl 0666.55007

Given a finite group G, equivariant rational homotopy theory provides, for any G system of Q-algebras (i.e., covariant functor \({\mathcal H}\) from coset G-spaces G/K to connected graded commutative Q-algebras of finite type), the existence of a G-CW-complex X (of finite type) with the property that \(H*(X^ K;Q)={\mathcal H}*(G/K)\), for any subgroup \(K\subset G\) [see G. Triantafillou, Proc. Am. Math. Soc. 89, 713-716 (1983; Zbl 0548.57023)]. In contrast to the non-equivariant case the uniqueness up to G-homotopy equivalence of such (rational) X does not hold in general, even in the most simple situations, namely for Moore G-spaces of type (\({\mathcal H},n)\), here \(\tilde {\mathcal H}*(G/K)\) is required to be concentrated in degree n for any K [as noticed by P. J. Kahn, Trans. Am. Math. Soc. 298, 245-271 (1986; Zbl 0616.55009)]. The paper under review takes a closer look at Kahn’s example and concludes the existence of infinitely many non-G-homotopy equivalent Moore G-spaces of type (\({\mathcal H},2)\).
Reviewer: St.Papadima

MSC:

55P91 Equivariant homotopy theory in algebraic topology
55P62 Rational homotopy theory
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