Quadratic functions in geometry, topology, and \(M\)-theory
type: Item
instance of: scholarly article
title: Quadratic functions in geometry, topology, and \(M\)-theory
zbMATH Open document ID: 1116.58018
author: Michael J. Hopkins
author: I. M. Singer
published in: Journal of Differential Geometry
publication date: 2006-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0211216
full work available at URL: https://projecteuclid.org/euclid.jdg/1143642908
review text: This paper provides a theory of quadratic functions that appear in global analysis (e.g., the Riemann parity) and algebraic topology (various quadratic refinements of intersection pairing) and is particularly concerned with the relationship between analytically and topologically defined quadratic functions. Here is a very brief review of some ideas and results of the paper. The review is necessarily far from being complete and misses many lovely ideas of the authors. The authors start with some examples of the (well understood) relationship between analytically and topologically defined quadratic functions that appear in the theory of Riemann surfaces and 4-dimensional manifolds, indicate obstructions to understand an analogous relationship in higher dimensions, and then provide a mathematical apparatus that enables them to formulate such a generalization. The first of the examples is provided by the index theorem for Spin\(^{c}\)-Dirac-type operators on a Spin\(^{c}\)-4-manifold \(M\) (i.e., a manifold the second Stiefel-Whitney class of which is the mod 2 reduction of an integral cohomology class) and a relationship between the index and intersection pairing. Thus let us recall that Spin\(^{c}\)-structures (if any) are in a 1-1 correspondence with \(H^{2}(M;\mathbb{Z})\), Spin\(^{c}\)-structures correspond to naturally defined complex line bundles (which are also classified by \(H^{2}(M;\mathbb{Z})\)) and an elliptic Dirac-type operator. Then, given a Spin\(^{c}\)-structure, the line bundle corresponding to this structure, and the first Chern class \(\lambda \) of the line bundle, the index theorem gives the following formula for the index \(\kappa (\lambda )\) (this notation is derived from surgery theory, in particular the theory of Kervaire-Arf invariants) of the corresponding Spin\(^{c}\)-operator: \(\kappa (\lambda )=\frac{1}{8}\int_{M}(\lambda ^{2}-L_{4k})\), where \(L_{4k}\) is the \(4k\)-th component of the Hirzebruch polynomial (and \(k=1\) in this instan…
Mathematics Subject Classification ID: 58J28
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 81T50
zbMATH DE Number: 5033792
zbMATH Keywords: generalized index theorem
zbMATH Keywords: differential cocycles, differential function spaces, differential cohomology theories
reviewed by: Wiesław Oledzki
MaRDI profile type: Publication
OpenAlex ID: W1851508906
arXiv ID: math/0211216
Wikidata QID: Q115197626
DOI: 10.4310/JDG/1143642908
description: scientific article