The Novikov conjecture. Geometry and algebra
Let \(M^n\) be a closed oriented manifold. It is well-known that Pontrjagin characteristic classes \(p_k(M)\in H^{4k}(M)\) are diffeomorphism invariant but not homotopy invariant in general. Nevertheless, some very special expressions containing \(p_k(M)\) are actually homotopy invariant. For example, if \(L(M)\in \bigoplus_{i\geq 0}H^{4i}(M)\) denotes the Hirzebruch \(L\)-class, which is a raional polynomial in the Pontrjagin classes, and if \(L_i(M)\in H^{4i}(M)\) is its \(i\)-th component then the famous Signature Theorem of Hirzebruch says that, for \(n=4k\), the signature of \(M\) equals the evaluation of \(L_k(M)\) on the fundamental class of \(M\), \(\langle L_k(M),[M]\rangle\), hence the latter is homotopy invariant. It is also known that a polynomial in the Pontrjagin classes is homotopy invariant only if is a multiple of \(L_k(M)\). Now consider the numbers \(\langle x\cup L_i(M),[M]\rangle\in\mathbb Q\) for \(x\in H^{n-4i}(M)\). They are also not homotopy invariant in general, but again some special choice of \(x\) gives us homotopy invariants. Let \(\pi=\pi_1(M)\) be the fundamental group of \(M\) and let \(B\pi\) denote the classifying space for \(\pi\). The numbers of the form \(\langle f^*(x)\cup L_i(M),[M]\rangle\in\mathbb Q\) for all \(f:M\to B\pi\) and all \(x\in H^{n-4i}(B\pi;{\mathbb Q})\) are called higher signatures. In the 1960s Novikov proved that all higher signatures are homotopy invariants for \(\pi=\mathbb Z\) and conjectured that the same is true for any fundamental group \(\pi\). The book under review originated from the lecture notes of the Oberwolfach Seminar on the Novikov conjecture and is written by leading experts as a guided tour to the Novikov conjecture. The lecture notes cover mostly topological and algebraic aspects of the conjecture, but the analytical side was not completely neglected. A number of exercises and hints to their solution is a great advantage of the book. It begins with basics about signature, higher signatures, Whitehead torsion and the \(s\)-cobordism theorem. Then an introduction to surgery theory is presented and the assembly map is defined and used to prove the Novikov conjecture for finitely generated free abelian groups. This part may be considered as an additional post-graduate course in topology. In the rest of the book some more recent important concepts and results on the Novikov conjecture and other closely related conjectures are presented. This starts with an introduction to spectra and classifying spaces of families, a generalization of aspherical spaces and continues with equivariant homology theories. Three subsequent chapters discuss two other important conjectures, the Farrell-Jones and the Baum-Connes ones, and their relation to the Novikov conjecture. In the final chapter the status of the conjectures is summarized and some computations in \(L\)-theory are presented.
scientific article; zbMATH DE number 2121589