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Infinite dimensional compacta having cohomological dimension two: An application of the Sullivan conjecture. (English) Zbl 0822.55001
In [Math. USSR, Sb. 63, No. 2, 539-546 (1989); translation from Mat. Sb., Nov. Ser. 135(177), No. 4, 551-557 (1988; Zbl 0643.55001)] A. N. Dranishnikov gave an example of an infinite dimensional metric compactum with integral cohomological dimension (\(\dim_ \mathbb{Z}\)) equal to 3. The authors produce an example with \(\dim_ \mathbb{Z} = 2\). One of the key points in Dranishnikov’s construction is that \(K(\mathbb{Z}, 3)\) is a point for \(K\)-theory with \(\mathbb{Z}/p\) coefficients. Since there is no “readily available” generalized cohomology theory for which \(K(\mathbb{Z}, 2)\) is a point the authors use the homotopy functor \([ , \Omega^ 3 S^ 3]\) and use the solution of the Sullivan conjecture [H. Miller, Ann. Math., II. Ser. 120, 39-87 (1984; Zbl 0552.55014)] to show that \([K(\mathbb{Z}^ n, 2),\;\Omega^ 3 S^ 3] = 0\). The construction then proceeds similar to the one by Dranishnikov using Edwards-Walsh complexes [J. J. Walsh, Lect. Notes Math. 870, 105-118 (1981; Zbl 0474.55002)].
Reviewer: E.Vogt (Berlin)

55M10 Dimension theory in algebraic topology
55S35 Obstruction theory in algebraic topology
55P35 Loop spaces
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