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basis boundary called Chapter choose classical clearly closed cohomology compact support components compute condition connected consider contained contravariant coordinate system coordinate system x,U COROLLARY countable covariant critical point define denote diagram diffeomorphism differential equations disjoint union equivalence classes exact example exp tY fibre finite function f functor geodesic hence Hint homeomorphic immersion implies inclusion map inner product integral curve integral manifold isomorphism k-form left invariant Lemma Let f Lie algebra Lie group manifold-with-boundary map f matrix metric space Mobius strip n-dimensional n-manifold notation obtain one-one open set open subsets order isomorphic orientation preserving oriented manifold Problem Proposition prove Riemannian metric satisfies sequence singular structure submanifold subspace Suppose tangent bundle tangent vector tensor field topology torus trivial unique vector bundle vector field vector space well-ordered set