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abelian adjoint functors assume BO(n bordism boundary canonical choose closed closed model category coalgebra cobordism cofibration coherent analytic sheaf cohomology commutative compact completes the proof Corollary critical points curve cycle defined denote dg coalgebra dg Lie algebra diagram diffeomorphic differential dimension element embedding equation equivariant exact sequence exists extended fibration filtration finite follows function functor given Hence Hilbert Hilbert manifold homology homotopy equivalence homotopy groups homotopy theory Hopf algebra hypersurface inclusion induced integral intersection invariant involution isomorphism isotopy kernel Lemma linear manifold Math matrix module morphism non-singular normal bundle obtained open subset Poincare pole of order polynomial proof of Theorem Proposition prove residual resp result ring satisfying simple covering simplicial simplicial sets smooth Stein open subalgebra subgroup Suppose surjective topological trivial tubular neighborhood unique vector space weak equivalence zero