Let $CW$ denote the collection of all finite CW complexes, and for some fixed integer $n$, consider a function $F:CW \to \Bbb Z$ satisfying the followings:
(1) If $X,Y \in CW$ are homotopy equivalent, then $F(X)=F(Y)$.
(2) If $X \in CW$ and $A$ is a subcomplex of $X$, then $F(X)=F(A)+F(X/A)$.
(3) $F(S^0)=n$.
I know that such a function $F$ exists, namely, $F(X)=n(\chi(X)-1) $, where $\chi$ is the Euler characteristic.
What I want to show is uniqueness, but I got stuck. How can I show that such an $F$ must be unique?
This is a nice question! If we can determine what the value of our function is on a wedge of two complexes we are basically done. This is not difficult. It follows from taking the subcomplex of the wedge to be one of the summands and applying the rule for quotients. The value on spheres is determined because by quotienting out the hemisphere we get an algebraic relationship we can solve for the value of the function on our sphere. It is also easy to figure out the value on n points inductively.
Then any finite complex can be obtained by inductively attaching disks to a complex that we already know the functions value on. So we may take $A$ to be the complex we are attaching to, and then we may compute $F(X)$ to be $F(A)$ plus the value on some sphere. Interestingly it doesn’t seem necessary to impose homotopy equivalence, only homeomorphism equivalence.