# What does go wrong in Cellular homology if one considers projective limits of celullar complexes instead of CW-complexes?

Consider a nice topological space $$X$$ (e.g. the 3-sphere) and consider inside a decreasing sequence of compact subsets $$(K_n)_{ninmathbb N}$$ such that $$K_infty:=bigcap_{nin mathbb N} K_n$$ is 0-dimensional. Assume further more that you have a filtration of CW-complexes $$(mathcal S_n)_{nin mathbb N}$$, each on $$Xsetminus Int(K_n)$$ for all $$nin mathbb N$$. Now consider the projective limit of the system
$$X/K_{n+1} xrightarrow{pi_{n+1,n}} X/K_n$$
where $$X/K_n := K/sim$$ with $$xsim y$$ identifying elements of the same connected components of $$K_n$$. We can thus define $$X’:=varprojlim X/K_n$$ which comes with a natural map $$Xrightarrow X’$$. Assuming $$X$$ and the $$K_n$$ are nice enough (like $$X$$ compact and for any neighborhood $$mathcal U$$ of any given $$xin K_infty$$ there exists an $$n$$ big enough so that the connected component of $$x$$ in $$K_n$$ is a subset of $$mathcal U$$), the map $$Xrightarrow X’$$ is an homeomorphism.

Now, one could try to define a " pro-cellular" homology for such a projective limit of CW-complexes : $$X^{(0)}$$ would be the 0-dimensional subset of $$X$$ given by all the vertices of the CW-complexes plus $$K_{infty}$$, then $$X^{(1)}$$ is given by all the edges of the CW-complexes,…

This cellular decomposition fails to be a CW-complex in several ways but the relative homology defining the cellular chain complex $$H(X^{(n)},X^{n-1};G)$$ is still composed of free $$G$$-modules and is what one would expect for a CW-complex i.e. the free module generated by the $$n$$-facets.

Does one still have equivalence with singular homology?, I guess the answer is No but I don’t see why.

MathOverflow Asked by Léo Brunswic on January 27, 2021

0 Answers

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