Dedekind sums, \(\mu\)-invariants and the signature cocycle. (English) Zbl 0809.11027

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.
Reviewer: R.Kirby (Berkeley)


11F20 Dedekind eta function, Dedekind sums
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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