The Borel/Novikov conjectures and stable diffeomorphisms of 4-manifolds
نویسنده
چکیده
Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S × S’s. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic provided a certain map from the second homology of the fundamental group with coefficients in Z2 to the L-theory of the group is injective. This injectivity is implied by the Borel/Novikov conjecture for torsion-free groups, which is known for many groups. There are also results concerning the homotopy invariance of the KirbySiebenmann invariant. The method of proof is to use Poincare duality in Spin bordism to translate between Wall’s classical surgery and Kreck’s modified surgery.
منابع مشابه
N ov 2 00 4 The Borel / Novikov conjectures and stable diffeomorphisms of 4 - manifolds
Two 4-dimensional manifolds M and N are stably diffeomorphic if for some nonnegative integers r and s, the connected sum M#r(S×S) is diffeomorphic to N#s(S× S). This sort of stabilization plays a fundamental role in 4-dimensional topology, see, for example [W1], [C-S]. In this paper we make the following two conjectures, relate them to standard conjectures in manifold theory, and thereby prove ...
متن کاملun 2 00 4 The Borel / Novikov conjectures and stable diffeomorphisms of 4 - manifolds
Two 4-dimensional manifolds M and N are stably diffeomorphic if for some nonnegative integers r and s, the connected sum M#r(S×S) is diffeomorphic to N#s(S× S). This sort of stabilization plays a fundamental role in 4-dimensional topology, see, for example [W1], [C-S]. In this paper we make the following two conjectures, relate them to standard conjectures in manifold theory, and thereby prove ...
متن کامل“L-invariants of regular coverings of compact manifolds and CW -complexes”
0. Introduction 1. L-Betti numbers for CW -complexes of finite type 2. Basic conjectures 3. Low-dimensional manifolds 4. Aspherical manifolds and amenability 5. Approximating L-Betti numbers by ordinary Betti numbers 6. L-Betti numbers and groups 7. Kähler hyperbolic manifolds 8. Novikov-Shubin invariants 9. L-torsion 10. Algebraic dimension theory of finite von Neumann algebras 11. The zero-in...
متن کاملLarge scale geometry, compactifications and the integral Novikov conjectures for arithmetic groups
The original Novikov conjecture concerns the (oriented) homotopy invariance of higher signatures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence related Novikov and i...
متن کاملClassification of Partially Hyperbolic Diffeomorphisms in 3-manifolds with Solvable Fundamental Group
A classification of partially hyperbolic diffeomorphisms on 3-dimensional manifolds with (virtually) solvable fundamental group is obtained. If such a diffeomorphism does not admit a periodic attracting or repelling two-dimensional torus, it is dynamically coherent and leaf conjugate to a known algebraic example. This classification includes manifolds which support Anosov flows, and it confirms...
متن کامل