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Cohomological dimension with respect to perfect groups. (English) Zbl 0873.55001

The authors consider the class of compacta with cohomological dimension one, with respect to perfect groups. In particular, they prove that there exist certain compacta, alled Cannon-Stan’ko compacta, of arbitrarily high dimension. A compactum \(X\) is said to be a Cannon-Stan’ko compactum provided that the cohomological dimension of \(X\) is at most one, with respect to the fundamental group of a minimal group, which is the direct limit of 2-polyhedra, which are constructed by attaching two 1-handles to each 1-handle pair of generators of the one-dimensional homology of the previous complex, and that the genus of the initial complex is one. Finally, they prove that the weakly Cainian compacta are precisely the two-dimensional Cannon-Stan’ko compacta.

MSC:

55M10 Dimension theory in algebraic topology
57Q55 Approximations in PL-topology
54F45 Dimension theory in general topology
54C25 Embedding
57Q65 General position and transversality
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