## Annals of Mathematics |

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2-sphere abelian group Amer analytic assume asymptotic Borel boundary bundle closure co-torsion cochain coefficients coherent sheaf cohomology commutative compact complex components condition consider contains Corollary corresponding countable cyclic groups defined definition denote diagram differential equations dimension dimensional direct sum disk dual dual spaces element exact sequence exists Ext QIZ factors of automorphy finite number follows Frobenius algebra function geodesic given Hence homeomorphism homology homotopy class Horn ideal identity implies induces integral curve intersection isomorphism Lemma linear locally polyhedral manifold Math matrix metric neighborhood obtain open set orthogonal pair periodic solution Pext polynomial positive projection proof of Theorem properties Proposition prove regular respect result Riemannian manifold satisfies semi-simplicial sheaf space subgroup subset subspace Suppose tangent Theorem tion topology torsion torsion-free trajectory triangulation trivial vector field weak homotopy equivalence zero