##
**On the Novikov conjecture.**
*(English)*
Zbl 0954.57017

Ferry, Steven C. (ed.) et al., Novikov conjectures, index theorems and rigidity. Vol. I. Based on a conference of the Mathematisches Forschungsinstitut Oberwolfach held in September 1993. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 226, 272-337 (1995).

From the introduction: Signatures of quadratic forms play a central role in the classification theory of manifolds. The Hirzebruch theorem expresses the signature \(\sigma(N)\in\mathbb{Z}\) of a \(4k\)-dimensional manifold \(N^{4k}\) in terms of the \({\mathcal L}\)-genus \({\mathcal L}(N)\in H^{4*} (N;\mathbb{Q})\). The ‘higher signatures’ of a manifold \(M\) with fundamental group \(\pi_1(M) =\pi\) are the signatures of the submanifolds \(N^{4k}\subset M\) which are determined by the cohomology \(H^*(B\pi; \mathbb{Q})\). The Novikov conjecture on the homotopy invariance of the higher signatures is of great importance in understanding the connection between the algebraic and geometric topology of high-dimensional manifolds. Progress in the field is measured by the class of groups \(\pi\) for which the conjecture has been verified. A wide variety of methods has been used to attack the conjecture, such as surgery theory, elliptic operators, \(C^*\)-algebras, differential geometry, hyperbolic geometry, bounded/controlled topology, and algebra.

The object of this largely expository paper is to outline the relationship between the Novikov conjecture, the exotic spheres, the topological invariance of the rational Pontryagin classes, surgery theory, codimension 1 splitting obstructions, the bounded/controlled topology of non-compact manifolds, the algebraic theory of A. A. Ranicki [Proc. Lond. Math. Soc., III. Ser. 40, 87-192 (1980; Zbl 0471.57010); 193-283 (1980; Zbl 0471.57011); Lower \(K\)- and \(L\)-theory, Lond. Math. Soc. Lect. Note Ser. 178 (1992; Zbl 0752.57002); Algebraic \(L\)-theory and topological manifolds, Camb. Tracts Math. 102 (1992; Zbl 0767.57002)], and the method used by G. Carlsson and E. K. Pedersen [Topology 34, No. 3, 731-758 (1995; Zbl 0838.55004)] to prove the conjecture for a geometrically defined class of infinite torsion-free groups \(\pi\) with \(B\pi\) a finite complex and \(E\pi\) a non-compact space with a sufficiently nice compactification. See [S. C. Ferry, A. A. Ranicki and J. Rosenberg, A history and survey of the Novikov conjecture, Lond. Math. Soc. Lect. Note Ser. 226, 7-66 (1995), see below] for a wider historical survey of the Novikov conjecture.

For the entire collection see [Zbl 0829.00027].

The object of this largely expository paper is to outline the relationship between the Novikov conjecture, the exotic spheres, the topological invariance of the rational Pontryagin classes, surgery theory, codimension 1 splitting obstructions, the bounded/controlled topology of non-compact manifolds, the algebraic theory of A. A. Ranicki [Proc. Lond. Math. Soc., III. Ser. 40, 87-192 (1980; Zbl 0471.57010); 193-283 (1980; Zbl 0471.57011); Lower \(K\)- and \(L\)-theory, Lond. Math. Soc. Lect. Note Ser. 178 (1992; Zbl 0752.57002); Algebraic \(L\)-theory and topological manifolds, Camb. Tracts Math. 102 (1992; Zbl 0767.57002)], and the method used by G. Carlsson and E. K. Pedersen [Topology 34, No. 3, 731-758 (1995; Zbl 0838.55004)] to prove the conjecture for a geometrically defined class of infinite torsion-free groups \(\pi\) with \(B\pi\) a finite complex and \(E\pi\) a non-compact space with a sufficiently nice compactification. See [S. C. Ferry, A. A. Ranicki and J. Rosenberg, A history and survey of the Novikov conjecture, Lond. Math. Soc. Lect. Note Ser. 226, 7-66 (1995), see below] for a wider historical survey of the Novikov conjecture.

For the entire collection see [Zbl 0829.00027].

### MSC:

57R67 | Surgery obstructions, Wall groups |

19J25 | Surgery obstructions (\(K\)-theoretic aspects) |

57R20 | Characteristic classes and numbers in differential topology |