Open AccessCompact negatively curved manifolds (of dim [unk] 3,4) are topologically rigid
Author(s) -
F. T. Farrell,
L. E. Jones
Publication year - 1989
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.86.10.3461
Subject(s) - homotopy , mathematics , riemannian manifold , homeomorphism (graph theory) , manifold (fluid mechanics) , pure mathematics , fundamental group , simply connected space , lie group , rigidity (electromagnetism) , equivalence (formal languages) , combinatorics , physics , topology (electrical circuits) , quantum mechanics , mechanical engineering , engineering
LetM be a complete (connected) Riemannian manifold having finite volume and whose sectional curvatures lie in the interval [c 1 ,c 2 ] with -∞ <c 1 [unk]c 2 < 0. Then any proper homotopy equivalenceh:N →M from a topological manifoldN is properly homotopic to a homeomorphism, provided the dimension ofM is >5. In particular, ifM andN are both compact (connected) negatively curved Riemannian manifolds with isomorphic fundamental groups, thenM andN are homeomorphic provided dimM [unk] 3 and 4. {If both are locally symmetric, this is a consequence of Mostow's rigidity theorem [Mostow, G. D. (1967)Publ. Inst. Haut. Etud. Sci. 34, 53-104].} WhenM has infinite volume we can still calculate the surgeryL -groups of π1 M , even when dimM = 3, 4, or 5, providedM is locally symmetric. An identification of the weak homotopy type of the homeomorphism group of (finite volume)M is also made through a stable range. We have previously announced these results for the special case thatc 1 =c 2 = -1.