What is the difference between the notion of cell decomposition of a CW-complex, and the notion of stratification of a topological space ?

I know that cell decomposition of a CW-complex is usefull to define cellular homology and cohomology. Which kind of homology and cohomology is defined by the use of the notion of stratification if by analogy, cell decomposition of a CW-complex is usefull to define cellular homology and cohomology ?

Thanks in advance for your help.

  • 1
    $\begingroup$ There are entirely too many notions of "stratification" (eg Whitney, Thom-Mather, Quinn,...) out there for there to be a good answer to your question; but at least in the special case of "cone-like stratified spaces", maybe the most important difference is that the "n-cells" don't have to look anything like closed disks, but rather can be arbitrary n-manifolds. $\endgroup$ – Vidit Nanda Aug 25 '18 at 17:12
  • $\begingroup$ Thank you. And, which kind of homology and cohomology we might construct with one of these kinds of stratification that you mentionned ? $\endgroup$ – YoYo Aug 25 '18 at 17:20
  • $\begingroup$ Notions of (co)homology require a differential map $d$ from one "stage" to the next, with the condition that $d^2=0$. The definitions of stratification don't usually come with any differential map between the strata; most often the only connecting morphism between strata is the inclusion map, which does not square to $0$. $\endgroup$ – Jānis Lazovskis Aug 25 '18 at 22:05
  • $\begingroup$ @Laziovskis But there is the spectral sequence for compactly supported cohomology, which generalizes the CW homology chain complex. $\endgroup$ – Phil Tosteson Aug 26 '18 at 1:07
  • $\begingroup$ One of the reasons that you can get a homology theory from the CW stratification is that $H_i(X^n,X^{n-1})$ is a direct sum of one $\mathbb Z$ for each $n$-cell in degree $n$ and $0$ otherwise. This is then identified as the degree $n$ chain group, generated by the $n$-cells. The connecting morphism of the long exact sequence of the triple can then be identified as a boundary map of the corresponding chain complex. This all tends to fail with more general stratifications, which is why other types of stratifications don't tend to be associated with homology theories. $\endgroup$ – Greg Friedman Aug 26 '18 at 4:13

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