What is the Euler characteristic in topology?

The Euler characteristic is a topological invariant, represented by the Greek letter chi, that describes a space's shape or structure regardless of how it is bent. Originally defined for polyhedra and Platonic solids, it now arises in modern mathematics through homology and homological algebra. Leonhard Euler introduced it for convex polyhedra, though he did not rigorously prove its invariant nature.

What we know

• The Euler characteristic is a topological invariant that describes a topological space's shape or structure regardless of bending. [1, 2]
• Francesco Maurolico stated the concept for Platonic solids in an unpublished manuscript in 1537. [2]
• Leonhard Euler introduced the concept for convex polyhedra more generally but failed to rigorously prove that it is an invariant. [2]
• In modern mathematics, the Euler characteristic arises from homology and homological algebra. [2]

Where to go next

• whoWho was Leonhard Euler?
• a case in pointWhat are Platonic solids?
• the bigger pictureWhat is algebraic topology?
• how it workedWhat is homology in mathematics?
• what followedWhat is the Euler class?
• an unexpected connectionHow does a tangent bundle relate to the Euler characteristic?The Euler class of a smooth manifold's tangent bundle generalizes the classical Euler characteristic to vector bundles.

Sources

• [1]Euler characteristic (Wikidata Q852973) · CC0 1.0
• [2]Euler characteristic · CC BY-SA 4.0
• [3]List of topics named after Leonhard Euler · CC BY-SA 4.0
• [4]Surface (topology) · CC BY-SA 4.0
• [5]Euler class · CC BY-SA 4.0