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Euler-Poincare formula

If a finite simplicial complex has

f_{i}

simplices of dimension

i

, then

χ (X) = \sum_{i} (- 1)^{i} f_{i} .

Writing

C_{i} = Z_{i} \oplus

a lift of

B_{i - 1}

and

Z_{i} = H_{i} \oplus B_{i}

makes the boundary dimensions cancel in the alternating sum.

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