Stable cohomotopy groups of compact spaces

Volume 180 / 2003

Sławomir Nowak Fundamenta Mathematicae 180 (2003), 99-137 MSC: Primary 55P25, 55P55, 55N20; Secondary 55N07. DOI: 10.4064/fm180-2-1

Abstract

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 $\pi^k_{s}(X)$.

Any cohomology functor on the homotopy category of compact spaces factorizes via the stable shape category $\rm ShStab$. 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 $\pi^n_s(X)=0$ 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 ${\mathbb W}_X$ assigned to $X$ and the class of all $X$ with ${\mathfrak C}^{s}(X)= \max \{k:\pi ^k_s(X) \neq 0\}<\infty$ corresponds to the class of spectra bounded below.

The notion of the cohomological dimension $\mathfrak {H}$-$\dim X$ with respect to a generalized cohomology theory ${\mathfrak {H}}$ is studied. In particular we show that $\boldsymbol{\pi}\hbox{-}\!\dim X \geq \mathfrak {H} \hbox{-}\!\dim X$ for every ${\mathfrak {H}}$ and $\boldsymbol{\pi}\hbox{-}\!\dim X = \infty $ if $\boldsymbol{\pi}\hbox{-}\!\dim X > \dim _\mathbb{Z} X,$ where $\boldsymbol{\pi}$ is the stable cohomotopy theory and $\dim _\mathbb{Z} X$ is the integral cohomological dimension. The following question remains open: does $ \boldsymbol{\pi}\hbox{-}\!\dim X $ coincide with $\dim X ?$

Authors

  • Sławomir NowakInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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