Singular $\simeq$ Cellular homology? Given an arbitrary CW-complex, are the singular chain complex $S_\ast(X)$ and cellular chain complex $C_\ast(X)$ homotopy equivalent or just quasi-isomorphic (some chain map induces isomorphisms on homologies) or only have isomorphic homologies?
I can't find this in the standard AlgTop books. Any references are welcome.
Related: https://mathoverflow.net/questions/59390/when-is-a-quasi-isomorphism-necessarily-a-homotopy-equivalence
 A: You will find this kind of result in 
Blakers, A. "Some relations between homology and homotopy groups". Ann. of Math. (2) 49 (1948) 428--461.
I am pretty sure it is in Massey's book on Singular Homology, from a cubical viewpoint. 
Proposition 14.7.1 of Nonabelian Algebraic Topology gives a deformation of the singular cubical complex of a space onto  that coming from a filtration, under conditions which are satisfied in the case of a cellular filtration. 
Later: Here is the detail of the proposition. For the question you can assume $X_*$ is the skeletal filtration of a CW-complex and  $R X_*$ is the cubical set of cellular maps $I^n_* \to X_*$: 
Let $X_*$ be a filtered
space such that the following conditions $\psi (X_*, m)$ hold for
all $m \geqslant 0$:


*

*$\psi (X_*, 0) :$ The map $\pi_0 X_0 \rightarrow
  \pi_0 X$ induced by inclusion is surjective;

*$\psi (X_*, 1) :$ 
Any path in $X$ joining points of $X_0$ is deformable in $X$ rel
end points to a path in $X_1$; 

*$\psi (X_*, m) (m \geqslant 2
 ) :$ For all $\nu \in X_0$ , the map
$$\pi_m (X_m , X_{m-1} , \nu ) \rightarrow \pi_m (X, X_{m-1} , \nu
)$$ induced by inclusion is surjective. 
Then the
inclusion $i  \colon  RX_* \rightarrow KX=S^\square X$ is a homotopy
equivalence of cubical sets.
The proof is quite direct by induction because the relative homotopy groups may be defined by maps of cubes, and in cubical sets, homotopies are defined using cubes.  
A: Here is a nice zig-zag of quasi isomorphisms. Let $Sing(X)$ denote the realization of the singular set of X. Let $Song(X)$ denote the realization of the simplicial set of singular simplices that are cellular maps.
We have a chain of maps $X \leftarrow Song(X) \rightarrow Sing(X)$, where it is standard that these are weak equivalences and by design are cellular (where the latter spaces are CW complexes since they are realizations of simplicial sets). Hence, on CW chains these are quasi isomorphisms.
It is easy to see that CW and simplicial homology on the realization of a simplicial set coincide, so after taking CW chains we can extend to the right by an isomorphism of chain complexes getting us the simplicial chains of $Sing(X)$. Simplicial chains on $Sing(X)$ is exactly singular chains on X, so we are done.
