A cancellation formula of Alvarez-Gaumé and Witten and a signature defect-type invariant of 11-dimensional lens spaces with spin structures

Yoshihiro Fukumoto

Vol. 235 (2008), No. 2, 213–234

DOI: 10.2140/pjm.2008.235.213

Abstract

In this paper, we define an integer-valued diffeomorphism invariant of 11-dimensional lens spaces with spin structures. This invariant modulo 16 gives the generalized Rohlin invariant and is defined in a way analogous to the signature defects by using a cancellation formula which discovered by L. Alvarez-Gaumé and E. Witten in their study of gravitational anomalies. In particular, we give an explicit formula for the invariant by using the Kawasaki V -index theorem, and we calculate the invariant for several examples of lens spaces. Using this formula, we obtain a necessary condition for smooth 11-dimensional free ℤ∕p-spheres to be the boundaries of 12-dimensional free spin ℤ∕p-manifolds. We also prove that this invariant has a reciprocity property similar to the reciprocity law of the Theta multiplier given by B. Berndt.

Keywords

index theorem, V -manifold, reciprocity law, generalized Rohlin invariant, lens space

Mathematical Subject Classification 2000

Primary: 58J28, 58J20, 11A15

Secondary: 55N22, 53C27

Milestones

Received: 21 May 2007

Revised: 22 September 2007

Accepted: 4 December 2007

Published: 1 April 2008