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The 37 pages that link to Euler characteristic, each with the reason it gives.

• TopologyRelated: It summarizes structural information that remains stable under suitable deformations.

• HomeomorphismRelated: For suitable spaces, this invariant helps distinguish topological types.

• Algebraic topologyRelated: This compact invariant summarizes homological information across dimensions.

• FullereneRelated: For a closed trivalent cage, it constrains the ring pattern to twelve pentagons.

• Tree (graph theory)Related: For a tree, the vertex count minus the edge count equals one.

• Vertex (geometry)Related: For convex polyhedra, vertex counts enter the relation V − E + F = 2.

• Gauss–Bonnet theoremRelated: The theorem’s curvature integral is fixed by this invariant.

• Homology groupRelated: It can be calculated as an alternating sum of the ranks of homology groups.

• TorusRelated: The torus has Euler characteristic zero, consistent with its genus.

• Morse theoryRelated: The alternating sum of Morse critical-point counts equals this invariant.

• TetrahedronRelated: Its polyhedral formula gives 4 − 6 + 4 = 2 for a tetrahedron.

• August Ferdinand MöbiusRelated: The strip’s topology can be distinguished from familiar surfaces through invariants such as this one.

• DiagonalRelated: Polyhedral diagonal counts interact with vertex and edge counts in combinatorial geometry.

• CohomologyRelated: Finite-dimensional cohomology groups recover this compact summary of a space's topology.

• Möbius stripRelated: The strip's Euler characteristic is zero, linking its boundary and topology.

• Poincaré–Hopf theoremRelated: The sum of zero indices is constrained to equal this global invariant.

• Handshaking lemmaRelated: Degree sums help relate vertices and edges in planar graph formulas.

• Lefschetz fixed-point theoremRelated: For the identity map, the Lefschetz number equals this topological invariant.

• Chern–Gauss–Bonnet theoremRelated: It is the topological quantity recovered by integrating the Euler form.

• IcosahedronRelated: Its formula checks the icosahedron's counts of vertices, edges, and faces.

• Regular dodecahedronRelated: Its counts satisfy 20 − 30 + 12 = 2.

• Grothendieck–Riemann–Roch theoremRelated: Taking the map to a point turns the theorem into a formula for Euler characteristics.

• Hairy ball theoremRelated: The even-dimensional sphere's Euler characteristic forces zeros through Poincaré–Hopf.

• Riemann–Hurwitz formulaRelated: Writing genus through Euler characteristic makes the formula’s correction transparent.

• Heawood conjectureRelated: Surface genus enters the formula through the topology encoded by Euler characteristic.

• Pick's theoremRelated: Cell-counting arguments account for the constant minus one in the area formula.

• Carathéodory conjectureRelated: A sphere's Euler characteristic underlies the topological obstruction behind the conjecture.

• Cell (topology)Related: Counting cells by dimension gives a direct way to calculate this invariant.

• Hadwiger's theoremRelated: It is the zero-dimensional intrinsic volume and one term in the theorem's expansion.

• Harnack's curve theoremRelated: Topology constrains the real components through invariants of the underlying curve.

• Honeycomb theoremRelated: Planar topology constrains how many sides cells can have on average.

• Zig-zag lemmaRelated: The long exact sequence can relate Euler characteristics across a short exact sequence of complexes.