Stable cohomotopy groups of compact spaces

Sławomir Nowak¹

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Fundamenta Mathematicae, Tome 180 (2003) no. 2, pp. 99-137.

Résumé

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 $\dim X ?$

Détail

DOI : 10.4064/fm180-2-1

Keywords: reduce study global particular cohomological properties compact hausdorff space study its stable cohomotopy groups cohomology functor homotopy category compact spaces factorizes via stable shape category shstab main reason why language technique stable shape theory describe analyze global structure compact spaces given hausdorff compact space there exists metric compact space stable shape stable cohomotopy groups countable almost integral cohomology groups countable respectively finitely generated k fold suspension has stable shape finite dimensional compact metric space respectively finite complex sufficiently large there duality between compact hausdorff spaces spectra under which stable cohomotopy groups correspond homotopy groups spectrum mathbb assigned class mathfrak max neq infty corresponds class spectra bounded below notion cohomological dimension mathfrak dim respect generalized cohomology theory mathfrak studied particular boldsymbol hbox dim geq mathfrak hbox dim every mathfrak boldsymbol hbox dim infty boldsymbol hbox dim dim mathbb where boldsymbol stable cohomotopy theory dim mathbb integral cohomological dimension following question remains does boldsymbol hbox dim coincide dim