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Hairy ball theorem: Quiz

  

Question 1: There is a closely-related argument from ________, using the Lefschetz fixed point theorem.
Abstract algebraMathematicsCategory theoryAlgebraic topology

Question 2: For each point p, construct the ________ of s(p) with p as the point of tangency.
Lambert azimuthal equal-area projectionStereographic projectionManifoldConformal map

Question 3: In the case of the ________, the Euler characteristic is 0; and it is possible to 'comb a hairy doughnut flat'.
Stereographic projectionTorusManifoldMöbius strip

Question 4: By integrating a vector field we get (at least a small part of) a one-parameter group of ________ on the sphere; and all of the mappings in it are homotopic to the identity.
ManifoldDiffeomorphismFiber bundleDifferentiable manifold

Question 5: If f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always ________ to the sphere at p, then there is at least one p such that f(p) = 0.
TangentDerivativeManifoldTrigonometric functions

Question 6: The hairy ball theorem of ________ states that there is no nonvanishing continuous tangent vector field on a sphere.
Algebraic topologyCategory theoryMathematicsAbstract algebra

Question 7: In this regard, it follows that for any compact regular 2-dimensional ________ with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.
GeometryDifferentiable manifoldManifoldDifferential geometry
















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