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Homology and the Euler characteristic

Counting holes in every dimension with chain complexes.

Homology groups H₀, H₁, H₂ count connected pieces, tunnels and voids; their alternating ranks give the Euler characteristic, the same V − E + F from graph theory. Picture it: the cube: 8 − 12 + 6 = 2, like the sphere it can be inflated into. Think it: the boundary of a boundary is zero, ∂² = 0, and that one identity generates the whole theory.

Worked example · 6 - 12 + 8

Arithmetic · Arithmetic

Evaluate 6 - 12 + 8

6 - 12 + 8

Step by step

  1. -12 + 6 + 8 = 2

    Add: -12 + 6 + 8 = 2.

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How to: Homology and the Euler characteristic

  1. Add: -12 + 6 + 8 = 2.

Questions people ask

What does the fundamental group measure?

Loops in a space up to deformation. In a plane every loop shrinks to a point (trivial group); around a hole a loop can wind n times (the integers).

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