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Polyhedra of piecewise constant curvature
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algebra analytic aspherical boundary Chern cohomology compact components computation Conjecture consider convex Corollary Coxeter Coxeter group curvature cycle decomposition defined definition Dehn invariant denote determinant line bundle dimension dimensional Dirac operator element elliptic singularity embedded equivalent Euclidean Euler characteristic exact sequence example exceptional divisor Exercise fibration finite flag complex formula four-manifold function fundamental group gauge geodesic geometric genus Gorenstein singularity hence holomorphic homeomorphic homology homotopy hyperbolic 3-manifolds implies integral intersection form isometry isomorphism JSJ decomposition Kahler Lemma line bundle Math matrix metric minimal resolution graph minimally elliptic moduli space multiplicity nonpositively curved orbifold oriented polytope proof prove quaternion quotient rational result riemannian manifold scissors congruence Seiberg-Witten simplex simply connected smooth solution spherical Spinc structure subgroup subset Suppose SW equations SW invariant symplectic tangent Theorem theory topological torus triangle trivial unique vector vertex vertices Z^in Zmin