Dedekind sums, \(\mu\)-invariants and the signature cocycle
type: Item
instance of: scholarly article
title: Dedekind sums, \(\mu\)-invariants and the signature cocycle
zbMATH Open document ID: 0809.11027
DOI: 10.1007/BF01459782
author: Robion C. Kirby
author: Paul Melvin
published in: Mathematische Annalen
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165207
review text: This paper gives a geometric formulation of the Rademacher \(\varphi\)- function and of Dedekind sums in terms of the action of the modular group on the hyperbolic plane. This approach leads to simple continued fraction formulas involving signatures, which illuminate the appearance of Dedekind sums in many signature related formulas in topology. Several of these are discussed in the paper, including formulas for the signature defects and \(\mu\)-invariants of lens spaces and torus bundles over the circle, and for the well known signature cocycle on \(\text{SL} (2,\mathbb{Z})\). Some consequences of this point of view include (1) the existence of naturally defined integer lifts of the \(\mu\)-invariants of lens spaces (related also to their Brown invariants), (2) new formulas for \(p\)-signatures of knots, and (3) a simple formula for the signature cocycle.
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 683142
zbMATH Keywords: Rademacher function
zbMATH Keywords: Dedekind sums
zbMATH Keywords: continued fraction
zbMATH Keywords: signatures
zbMATH Keywords: signature defects
zbMATH Keywords: \(\mu\)-invariants
zbMATH Keywords: lens spaces
zbMATH Keywords: torus bundles
zbMATH Keywords: signature cocycle
zbMATH Keywords: Brown invariants
zbMATH Keywords: \(p\)-signatures of knots
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 683142