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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.

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    You ask for a detailed reference. Do you know of any references that mention this perspective, maybe not in detail? I haven't seen it before. It seems to me that you have attaching maps $S^{n-1} \to X^{n-1}/X^{n-2}$ and you probably just want to suspend these and wedge them together (and identify the domain with $X^n/X^{n-1}$). I think this is the same as what you wrote? The fact that you get a differential should be checked just as in the case of the ordinary cellular boundary map, e.g. in Hatcher's book. Does this not work?2012-06-14
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    This formulation is used in the paper "Equivariant ordinary homology and cohomology" by Costenoble and Waner, http://arxiv.org/pdf/math/0310237 to define equivariant cellular chains and is handled as if it were common knowledge. My motivation to use this boundary stems from the fact that I try to avoid orientation assumptions on the cells. In Bredons book, the cellular boundary is shown to be a boundary using mapping degrees and therefore orientation theory. I think (gonna check this), that Hatcher does the same thing. I am trying to understand this purely from the stable homotopy viewpoint.2012-06-15
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    Unfortunately I can't edit my comment any more... You find the definition on page 14 of the named paper.2012-06-15
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    Why are you using stable homotopy theory, though? The suspension homomorphisms $\pi_n(S^n) \rightarrow \pi_{n+1}(S^{n+1})$ are isomorphisms for $n\geq 1$, so really except for the bottom case there's no reason to introduce this language.2012-06-15
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    Of course, without equivariance one could replace stable homotopy groups by homotopy groups, but this does not help with the problem, does it? My main problem is to figure out why the map yields a boundary and how one can actually compute that boundary in terms of generators (without going back to the classical cellular description).2012-06-16

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