Show that for each \(n \in \mathbb{Z}\) there is a unique function \(\varphi\) assigning an integer to each finite CW complex, such that (a) \(\varphi(X) = \varphi(Y)\) if \(X\) and \(Y\) are homeomorphic, (b) \(\varphi(X) = \varphi(A) + \varphi(X/A)\) if \(A\) is a subcomplex of \(X\), and (c) \(\phi(S^0) = n\). For such a function \(\varphi\), show that \(\varphi(X) = \varphi(Y)\) if \(X \simeq Y\).
Solution
The unique function is \[ \boxed{\varphi(X)=n\,\widetilde\chi(X)=n(\chi(X)-1)} \] for every nonempty finite CW complex \(X\).
1. The axioms force \(\varphi(\mathrm{pt})=0\).
Take \(X=S^0\) and let \(A\) be one of its two points. Then \(X/A\cong S^0\), so \[ n=\varphi(S^0)=\varphi(\mathrm{pt})+\varphi(S^0). \] Hence \(\varphi(\mathrm{pt})=0\).
2. The axioms force \(\varphi(D^k)=0\) for every \(k\ge1\).
Choose a CW decomposition of \(D^k\) into two closed \(k\)-balls \(A\) and \(B\) meeting along a \((k-1)\)-ball, with \(A\) a subcomplex. Collapsing \(A\) to a point turns the other half-ball into another \(k\)-ball, so \[ D^k/A\cong D^k. \] By homeomorphism invariance and additivity, \[ \varphi(D^k)=\varphi(A)+\varphi(D^k/A)=2\varphi(D^k), \] hence \(\varphi(D^k)=0\).
3. For every \(k\ge0\),\[ \varphi(S^k)=(-1)^k n. \]
The case \(k=0\) is axiom (c). For \(k\ge1\), regard \(S^{k-1}=\partial D^k\) as a subcomplex of \(D^k\). Since \[ D^k/S^{k-1}\cong S^k, \] additivity and 2 give \[ 0=\varphi(D^k)=\varphi(S^{k-1})+\varphi(S^k). \] Thus \(\varphi(S^k)=-\varphi(S^{k-1})\), proving the formula inductively.
4. A finite discrete space of \(r\ge1\) points has value\[ (r-1)n. \]
The case \(r=1\) is 1 and \(r=2\) is axiom (c). For \(r>2\), choose a two-point subcomplex \(A\cong S^0\). Collapsing \(A\) to one point leaves a discrete space of \(r-1\) points, so induction and additivity give \[ \varphi(r\text{ points})=n+(r-2)n=(r-1)n. \]
5. For every finite CW complex \(X\),\[ \varphi(X)=n\widetilde\chi(X). \]
Let \(c_k\) be the number of \(k\)-cells. By 4, \[ \varphi(X^0)=(c_0-1)n. \] For \(k\ge1\), collapsing \(X^{k-1}\) gives \[ X^k/X^{k-1}\cong\bigvee^{c_k}S^k. \] Repeated additivity for wedge summands, together with 1 and 3, yields \[ \varphi(X^k/X^{k-1})=c_k(-1)^k n. \] Induction over the skeleta therefore gives \[ \varphi(X) =n\left((c_0-1)+\sum_{k\ge1}(-1)^kc_k\right) =n(\chi(X)-1). \] This proves uniqueness.
6. The formula \(\varphi(X)=n\widetilde\chi(X)\) satisfies axioms (a)–(c).
Homeomorphism invariance is clear. For a CW pair \((X,A)\), \[ \widetilde\chi(X)=\widetilde\chi(A)+\widetilde\chi(X/A), \] which follows immediately from cell counts: the quotient has the cells of \(X-A\) plus the quotient basepoint. Finally \[ \widetilde\chi(S^0)=1, \] so \(\varphi(S^0)=n\). Thus existence holds.
7. The function \(\varphi\) is a homotopy invariant.
For finite CW complexes, \[ \chi(X)=\sum_i(-1)^i\dim_{\mathbb Q}H_i(X;\mathbb Q), \] so Euler characteristic is preserved by homotopy equivalence. Therefore the reduced Euler characteristic, and hence \(\varphi=n\widetilde\chi\), is also preserved. Thus \[ X\simeq Y\quad\Longrightarrow\quad\varphi(X)=\varphi(Y). \]