Can cellular homology be defined without using simplicial or singular homology? One obstruction is to compute degree of attaching maps without invoking any homology theory.

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    $\begingroup$ The degree of the attaching maps makes sense without homology, through the homotopy groups. But defining cellular homology that way, although computationally useful, would make you lose a great thing about homology : it is a functor, that is, you can study how maps behave under it, and that's much better than just having the groups $\endgroup$ – Max Jun 4 at 10:56
  • $\begingroup$ Degree also makes sense using differential topology. $\endgroup$ – Lee Mosher Jun 4 at 13:25
  • $\begingroup$ I think the only difficult part would be showing that $\partial^2 =0$. Everything else is pretty straightforward. $\endgroup$ – Connor Malin Jun 4 at 15:11
  • $\begingroup$ Actually, the chain map condition as well might be hard. But I think there might be a way to prove $\partial ^2=0$ from the chain map condition by constructing an appropriate CW complex. $\endgroup$ – Connor Malin Jun 4 at 15:55

This is done in the book partially titled Nonabelian Algebraic Topology, (EMS $2011$), for which some background is in this paper. The main tool is a homotopically defined functor $\Pi: \text{Filtered Spaces} \to \text{Crossed Complexes} $ defined using fundamental groupoids $C_1= \pi_1(X_1,X_0)$ and relative homotopy groups $C_n =\pi_n(X_n, X_{n-1},X_0)$, and operations of the fundamental groupoids on the relative homotopy groups, and boundary maps $C_n \to C_{n-1}, n \geqslant 2$ (due to Blakers, 1948, for the case $X_0$ is a singleton). Thus an element of $C_n$ is a certain homotopy class of maps. In particular. when the filtered space $X_*$ is the skeletal filtration of a CW-complex, we get a strengthening of the usual cellular chain complex.

To make calculations, one needs a Higher Homotopy Seifert-Van Kampen Theorem for $\Pi$, which is proved by cubical higher groupoid methods. So it is complicated, but the original two papers, (RB and P.J. Higgins, JPAA, 1981) take just 61 pages, and do more than the classical cellular theory, particularly in dimension 2.

Of course the book includes much more than those papers, particularly tensor products, homotopies, and relations with chain complexes with operators.


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