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Exotic smooth structures on nonpositively curved symmetric spaces. (English) Zbl 0998.53033
Summary: We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the change of the smooth structure we use a tangential map from the locally symmetric space to its dual of compact type.

##### MSC:
 53C35 Differential geometry of symmetric spaces 57R55 Differentiable structures in differential topology 57R99 Differential topology 57T15 Homology and cohomology of homogeneous spaces of Lie groups 55R37 Maps between classifying spaces in algebraic topology
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##### References:
 [1] C S Aravinda, F T Farrell, Rank 1 aspherical manifolds which do not support any nonpositively curved metric, Comm. Anal. Geom. 2 (1994) 65 · Zbl 0840.53026 [2] W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics 61, Birkhäuser (1985) · Zbl 0591.53001 [3] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347, Springer (1973) · Zbl 0285.55012 [4] P Eberlein, Rigidity of lattices of nonpositive curvature, Ergodic Theory Dynam. Systems 3 (1983) 47 · Zbl 0521.53045 [5] F T Farrell, L E Jones, Negatively curved manifolds with exotic smooth structures, J. Amer. Math. Soc. 2 (1989) 899 · Zbl 0698.53027 [6] F T Farrell, L E Jones, Topological rigidity for compact non-positively curved manifolds, Proc. Sympos. Pure Math. 54, Amer. Math. Soc. (1993) 229 · Zbl 0796.53043 [7] F T Farrell, L E Jones, Complex hyperbolic manifolds and exotic smooth structures, Invent. Math. 117 (1994) 57 · Zbl 0804.53055 [8] M A Kervaire, J W Milnor, Groups of homotopy spheres I, Ann. of Math. $$(2)$$ 77 (1963) 504 · Zbl 0115.40505 [9] R C Kirby, L C Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies 88, Princeton University Press (1977) · Zbl 0361.57004 [10] G D Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton University Press (1973) · Zbl 0265.53039 [11] B Okun, Nonzero degree tangential maps between dual symmetric spaces, Algebr. Geom. Topol. 1 (2001) 709 · Zbl 1066.53100
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