Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible

Abstract

Abstract. In this paper we prove that for all n = 4k − 2, k ≥ 2 there exists a closed smooth complex hyperbolic manifold M with real dimension n having non-trivial π1(T <0 (M )). T <0 (M ) denotes the Teichmüller space of all negatively curved Riemannian metrics on M , which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity.

On negatively curved bundles with hyperbolic fibers outside the Igusa stable range

“…More precisely, they prove that the inclusion map F : T <0 (M ) → T (M ) (which forgets the negatively curved structure) is in general homotopically nontrivial. Similar results are obtained in [Sor14,FS17] when M is a Gromov-Thurston manifold or a complex hyperbolic manifold. These results can be translated in the language of bundle theory (c.f.…”

“…The Teichmüller space T (M ) = BDiff 0 (M ) classifies equivalence classes of smooth bundles with abstract fiber M and equipped with a fiber homotopy trivialization, whereas T <0 (M ) classifies equivalence classes of negatively curved bundles with abstract fiber M and equipped with a fiber homotopy trivialization. It follows from the results obtained in [FO09,Sor14,FS17] that there are negatively curved bundles which are nontrivial as smooth bundles with abstract fiber a hyperbolic manifold, a Gromov-Thurston manifold or a complex hyperbolic manifold. However they all represent elements of finite order in the homotopy groups of T (M ).…”

On the topology of moduli spaces of non-negatively curved Riemannian metrics

“…Especially in recent years there has been intensive activity and substantial further progress on these issues, compare, for example, [2,3,[6][7][8][9][10][11][13][14][15][16][17][18][20][21][22][25][26][27][28][29][30][31][33][34][35][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]58,60,61,[65][66][67][69]…”