The Novikov conjecture. Geometry and algebra
type: Item
instance of: scholarly article
title: The Novikov conjecture. Geometry and algebra
zbMATH Open document ID: 1058.19001
author: Wolfgang Lück
author: Matthias Kreck
published in: Oberwolfach Seminars
publication date: 2004-12-10T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 19J25
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 2121589
zbMATH Keywords: higher signatures
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: \(L\)-theory
zbMATH Keywords: surgery
zbMATH Keywords: assembly map
reviewed by: Vladimir M. Manuilov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2121589