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2003 | 180 | 2 | 99-137
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Stable cohomotopy groups of compact spaces

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We show that one can reduce the study of global (in particular cohomological) properties of a compact Hausdorff space X to the study of its stable cohomotopy groups $π^{k}_{s}(X)$.
Any cohomology functor on the homotopy category of compact spaces factorizes via the stable shape category ShStab. This is the main reason why the language and technique of stable shape theory can be used to describe and analyze the global structure of compact spaces.
For a given Hausdorff compact space X, there exists a metric compact space with the same stable shape iff the stable cohomotopy groups of X are countable. If $πⁿ_s(X) = 0$ for almost all n > 0 and the integral cohomology groups of X are countable (respectively finitely generated) for all n, then the k-fold suspension of X has the same stable shape as a finite-dimensional compact metric space (respectively a finite CW complex) for sufficiently large k.
There is a duality between compact Hausdorff spaces and CW spectra under which stable cohomotopy groups of X correspond to homotopy groups of the CW spectrum $𝕎_{X}$ assigned to X and the class of all X with $ℭ^{s}(X) = max{k: π^{k}_s(X) ≠ 0} < ∞$ corresponds to the class of spectra bounded below.
The notion of the cohomological dimension ℌ - dim X with respect to a generalized cohomology theory ℌ is studied. In particular we show that π - dim X ≥ ℌ - dim X for every ℌ and π - dim X = ∞ if $π - dim X > dim_{ℤ}X$, where π is the stable cohomotopy theory and $dim_{ℤ}X$ is the integral cohomological dimension. The following question remains open: does π - dim X coincide with dim X?
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  • Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm180-2-1
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