TITLE: Is there a homology theory that gives a *necessary and sufficient* condition for homotopy equivalence?
QUESTION [8 upvotes]: Is there a (non-Abelian) homology theory that realizes the following:

Let $X,Y$ be manifolds with complexes $C(X),C(Y)$.
  Then $X$ and $Y$ are homotopy-equivalent if and only if $C(X)$ and $C(Y)$ are isomorphic. 

Or maybe, the following? ("You need the map...")

Let $X,Y$ be manifolds with complexes $C(X),C(Y)$. Let $f:X\to Y$ be a continuous map which induces the "chain map" $f_*:C(X)\to C(Y)$.
  Then $f$ is a homotopy equivalence (it admits a $g:Y\to X$ such that $fg$ and $gf$ are homotopic to the identities) if and only if $f_*$ is an isomorphism.

If yes: which one/ones? An introduction to such theory?
If no: not yet, or is it impossible? Why?
No restriction for the objects in the complexes, they can be groups, modules, groupoids, anything else.
Of course, $X,Y$ may be required to be "nice enough" for the above to work. So, please state also the technical requirements. Moreover:

Same questions, but with cohomology instead of homology?

REPLY [6 votes]: What you could be looking for is Theorem 6 of J.H.C. Whitehead's "Combinatorial Homotopy II" Bull. Amer. Math. Soc. 55 (1949) 453-496. For a CW-complex K he defines what he calls a "homotopy system" $\rho (K)$  and the theorem says that for a map $f:K \to L$ of CW-complexes, $f$ is an equivalence if and only if $\rho(f)$ is an equivalence. However to get "only if $\rho(f)$ is an isomorphism" is unlikely for this functor.  
We have come to call $\rho(K)$ the fundamental crossed complex $\Pi(K_*)$ of the filtered space $K_*$, the filtration being by skeleta. Many aspects of Whitehead's work are dealt with in the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tract Vol 15, 2011. 
To add to the above, a recent preprint available from Graham Ellis' preprint page at Galway
G. Ellis and Le Van Luyen, "Homotopy 2-types of low order", preprint. 
lists isomorphism classes and quasi-isomorphism classes of some crossed modules of low order. 
This answer is related to my answer to this mathoverflow question.