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adjoint associated Atiyah Atiyah-Singer operator canonical carries Chern class Clifford algebra Clifford multiplication cobordism cohomology commutes compact manifold compact oriented compact spin manifold complex structure complex vector bundle connection consider construction Corollary Ct(X decomposition defined DEFINITION denote differential operator Dirac bundle Dirac operator element elliptic operator embedding equation equivalent example fact fibre finite formula fundamental Furthermore geometry given gives graded Hence hermitian homomorphism homotopy Index Theorem induced integral isomorphism K-theory Kahler Lemma Lie group line bundle manifold of dimension module non-zero normal bundle Note orthogonal orthonormal basis positive scalar curvature principal G-bundle principal symbol Proposition prove pure spinor quaternionic Recall representation result riemannian manifold riemannian metric scalar curvature self-adjoint smooth spin manifold spin structure Spin7 Spinn spinor bundle subgroup subspace Suppose tangent tensor product topological trivial vector bundle vector fields Z2-graded zero
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