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A-module abelian group abelian varieties algebraic group analytic assume automorphism beginning sequence biregular Blaschke product bounded C*-algebra closure cobordism cohomology commutative compact completes the proof conjugacy classes conjugate locus construction contains converges Corollary corresponding cyclic defined definition denote dimension direct sum elliptic complex endomorphisms equation equivalent exact sequence exists an element expm extended factor finite fixed point follows formula free abelian group function hence homology homomorphism homotopy hyperfinite implies induces integral ideal involution isomorphism kernel Lemma Let G Lie algebra linear link type manifold Math matrix module monomorphism morphism nilpotent non-zero notation obtain Poincare complex prime projective proof-figure Proposition prove rational representation resp restriction result ring satisfies semi-isolated semi-simple simply connected smooth Spin subgroup subset Suppose surjective topology trivial uhf algebra unipotent unitary upper sequence variable vector bundle vector space Zariski dense zero