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Euler characteristic: Quiz

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Question 1: In this setting, the Euler characteristic of a finite group or monoid G is 1/|G|, and the Euler characteristic of a finite ________ is the sum of 1/|Gi|, where we picked one representative group Gi for each connected component of the groupoid.
Category theoryGroup actionGroupoidVector space

Question 2: Multiple proofs, including their flaws and limitations, are used as examples in Proofs and Refutations by ________.
Alfred North WhiteheadImre LakatosAristotleKarl Popper

Question 3: The Euler characteristic can be defined for connected ________ by the same VE + F formula as for polyhedral surfaces, where F is the number of faces in the graph, including the exterior face.
Planar graphPlanarity testingGraph theoryGraph (mathematics)

Question 4: A version used in ________ is as follows.
MathematicsAlgebraic geometryNumber theoryCalculus

Question 5: if X is a ________, and one uses Euler characteristics with compact supports, no assumptions on M or N are needed.
Locally compact spaceBaire spaceDense setTopological space

Question 6: In mathematics, and more specifically in ________ and polyhedral combinatorics, the Euler characteristic (or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
Algebraic topologyCategory theoryAbstract algebraGeometry

Question 7: The Euler class, in turn, relates to all other ________ of vector bundles.
Pontryagin classCharacteristic classCohomologyChern class

Question 8: For closed smooth manifolds, the Euler characteristic coincides with the Euler number, i.e., the ________ of its tangent bundle evaluated on the fundamental class of a manifold.
Chern classEuler classCharacteristic classPontryagin class

Question 9: Another generalization of the concept of Euler characteristic on manifolds comes from ________.
Group (mathematics)GeometryOrbifoldFundamental group

Question 10: The concept of Euler characteristic of a bounded finite poset is another generalization, important in ________.
CalculusCombinatoricsDiscrete mathematicsMathematics







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