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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)

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