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Finiteness properties of self-equivalence groups of rational \(co\)-\(H\)- spaces. (English) Zbl 0762.55011

Given a nilpotent complex \(X\) which is rationally a bouquet of spheres, a co-\(H_ 0\)-space, the author investigates finiteness properties involving the group \(E(X)\) of the self-homotopy equivalences of \(X\) and the canonical subgroups \(E_ \pi(X)\) and \(E_ H(X)\), self-equivalences inducing the identity in homotopy and homology respectively. Besides, if \(X\) is 1-connected, explicit formulae for the rank of \(E_ H(X)\) and \(E(X)\) in terms of the rational homotopy and homology of \(X\) are displayed. The author also shows through a list of examples the necessity of \(X\) being finite and that \(E_ \pi(X)\) is not always finite for non co-\(H_ 0\)-spaces.
Reviewer: F.Gomez (Malaga)

MSC:

55P62 Rational homotopy theory
55P10 Homotopy equivalences in algebraic topology
55P45 \(H\)-spaces and duals
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