one can define cellular homology by letting $C_n(X)=\{\mathbb{S}^n, X^n/X^{n-1}\}$, where $X$ is a CW complex and the curly brackets mean stable homotopy classes of maps. Now the differential of the resulting complex is supposed to be given by a map $X^n/X^{n-1}\to\Sigma X^{n-1}/X^{n-2}$ and I struggle to understand this map. The only candidate I can think of would be the suspension of an attaching map. More precisely, on an $n$-cell, we use the inverse of the characteristic map to end up in $\mathbb{S}^n$, identify this with $\Sigma\mathbb{S}^{n-1}$ via a (canonical) homeomorphism, then apply the attaching map to end up in $\Sigma X^{n-1}$. If this is correct (is it?), I still do not see why this is indeed a differential, i.e. d^2=0.
Furthermore, I would like to do concrete calculations with this formulation, if necessary only in very low dimensions (say, 1 or 2). So in particular I would like to know how the ordinary cellular boundary operator can be recovered from the fancy one above. Does anyone know a detailed reference for this view on cellular homology or can provide any useful insights?
Thank you.