# Formula for Euler characteristic for quotient space of a CW complex

I know that there is formula for Euler characteristic:

$$\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B)$$

Is there any formula that links between (for CW complex) some complex, subcomplex and quotient complex ($\chi(X)$,$\chi(A)$,$\chi(X/A)$)?

I appreciate any help.

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What did you try? Did you work out some examples? Did you try the same method as for the union? –  studiosus Oct 15 '13 at 14:17

## 1 Answer

$\chi(X)=\chi(A)+\chi(X,A)$, see A. Dold, Lectures on Algebraic Topology, Chapt.V, Prop.5.7.

See also the remark of studiosus below.

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You are missing $-1$ in RHS. –  studiosus Oct 15 '13 at 15:37
@studiosus: Sorry, I don't understand you. –  Boris Novikov Oct 15 '13 at 15:50
Boris: $\chi(X/A)=\chi(X\cup_A CA)= \chi(X) - \chi(A) + \chi(CA)= \chi(X) -\chi(A) +1$. Here $CA$ is the cone over $A$. Thus, $\chi(X)= \chi(X/A)+ \chi(A) -1$. –  studiosus Oct 15 '13 at 16:21
@studiosus: Thank you, I corrected the answer. –  Boris Novikov Oct 15 '13 at 17:39