Can someone help me of how to prove these two basic properties of Euler characteristics, but regarding finite $CW$ complexes.
$a)$ If $A$ and $B$ are two subcomplexes of a finite $CW$ complex $X$, then
$\chi(X)=\chi(A)+\chi(B)-\chi(A \cap B)$
$b)$If $A$ is a subcomplex of a finite $CW$ complex $X$, then $\chi (A)-\chi (X)+\chi (X/A)=1$
Yes, I have the definition, which says that for $X$ being a finite $CW$ complex, of dimension n, and for $a_i$ being the number of $i$-cells of $X$, we define the Euler characteristic as the alternating sum
$\chi (X)=a_0-a_1+a_2-…+(-1)^na_n$,
but I find this cell_consideration a bit abstract for rather direct calculation which it seems should be used here.
Any help is very welcome!
Best Answer
Part (a): You could apply the inclusion-exclusion principle to each of the $a_i$'s individually. In other words, the number of $0$-cells in $X$ is equal to the number of $0$-cells in $A$ plus the number of $0$-cells in $B$ minus the number of $0$-cells in $A \cap B$. The same is true for $1$-cells, and $2$-cells, and so on: $$ a_0(X) = a_0(A) + a_0(B) - a_0(A \cap B), \\ a_1(X) = a_1(A) + a_1(B) - a_1(A \cap B), \\ a_2(X) = a_2(A) + a_2(B) - a_2(A \cap B), \\ \vdots$$
Part (b): When you quotient $X$ by $A$, you are shrinking $A$ to a point. That means you are replacing all the cells in $A$ with a single $0$-cell: $$ a_0 (X/A) = a_0(X) - a_0(A) + 1, \\ a_1(X / A) = a_1(X) - a_1(A) , \\ a_2(X / A) = a_2(X) - a_2(A), \\ \vdots $$