The Euler-Poincaré formula establishes a fundamental relationship between the Euler characteristic of a topological space and its homology groups. Specifically, it states that for a finite simplicial complex, the Euler characteristic is equal to the alternating sum of the ranks of the homology groups: $$ ext{χ} = ext{rank}(H_0) - ext{rank}(H_1) + ext{rank}(H_2) - ext{rank}(H_3) + ...$$ This formula highlights how the topology of a space can be quantified through algebraic invariants derived from its simplicial structure.
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