Lectures on surgical methods in rigidity
The Borel Conjecture asserts that for two closed aspherical manifolds \(M\) and \(N\) with isomorphic fundamental groups, there exists a homeomorphism \(f:M\to N\) inducing the given isomorphism of the fundamental groups. The Generalized Borel Conjecture asserts that \(|{\mathcal S} (M,\partial) |=1\) for a compact aspherical manifold \(M\) with boundary \(\partial M\). Here, \({\mathcal S} (M,\partial)\) is the structure set of the equivalence classes of pairs \((N,f)\), where \(N\) is a compact manifold with boundary \(\partial N\), \(f:(N,\partial N)\to (M, \partial M)\) is a homotopy equivalence which restricts to a homeomorphism \(f|_{\partial N}: \partial N\to \partial M\), and two pairs \((N,f)\) and \((N',f')\) are equivalent if there exists a homeomorphism \(h:(N, \partial N) \to(N', \partial N')\) such that \(f'\circ h\) is homotopic to \(f \text{ rel } \partial N\). The book under review originated from two courses given by the author at the Tata Institute of Fundamental Research. The book gives an exposition of the Farrell-Jones topological rigidity theorem (originally proven in [Proc. Symp. Pure Math. 54, Part 3, 229-274 (1993; Zbl 0796.53043)] that reads as follows. For a closed non-positively curved \(m\)-dimensional Riemannian manifold \(M\), \(|{\mathcal S} (M\times D^n, \partial) |=1\) when \(m+n \geq 5\) (Theorem 14.1). The result gives a partial verification of the Borel Conjecture and the Generalized Borel Conjecture. The proof uses a surgical method for analyzing \({\mathcal S} (M \times D^n, \partial)\). As ingredients, the method involves a result on an assembly map \([M \times D^n, \partial;G/ \text{Top}] \to L^2_{m+n} (\pi_1M)\) (Theorem 7.1), a result on vanishing of the Whitehead group \(\text{Wh} (\pi_1M)\) (Theorem 14.2), and a surgery result describing a condition under which a map \(f:N \to M\) is homotopic to a homeomorphism (Proposition 14.6). The exposition of the rigidity theorem is given in Lectures 1-14 of the book. Background material on related surgery theory and controlled topology is included. Lectures 15-20 of the book concern the question of smooth rigidity for nonpositively curved manifolds. Lecture 15 is devoted to the question of which closed smooth manifolds \(M\) support expanding endomorphisms. The lecture recalls a construction due to Farrell-Jones of an expanding endomorphism \(f:T^n\# \Sigma^n \to T^n\# \Sigma^n\) of the connected sum of the \(n\)-torus \(T^n\) \((n>4)\) and an arbitrary homotopy sphere \(\Sigma^n\) (Theorem 15.4). The construction shows that a manifold \(M\) supporting an expanding endomorphism though it is homeomorphic to an infranilmanifold (Theorem 15.3), it need not be diffeomorphic to an infranilmanifold. Lectures 16-19 contain some of the counterexamples to smooth rigidity due to Farrell-Jones. Lectures 16 is concerned with the problem of detecting nondiffeomorphic smooth structures on the same topological manifold. Lecture 17 is devoted to showing that for two closed smooth Riemannian manifolds \(M\) and \(N\) both negatively curved, \(\pi_1M \cong \pi_1N\) does not imply that \(M\) and \(N\) are diffeomorphic. This counterexample for smooth rigidity is obtained when \(M\) is a real hyperbolic manifold. Lectures 18 and 19 contain counterexamples for smooth rigidity when \(M\) is a complex hyperbolic manifold. Lecture 20 is a brief discussion (with references) to some recent results on topological and smooth rigidity, as well as PL rigidity.
non-positively curved Riemannian manifold
scientific article; zbMATH DE number 1043672