# A cell decomposition of a CW-complex and, stratification of a topological space

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 ?

• 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 $0$. – Jānis Lazovskis Aug 25 '18 at 22:05
• 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. – Greg Friedman Aug 26 '18 at 4:13