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Quantum invariants of periodic three-manifolds

Patrick M Gilmer

Geometry & Topology Monographs 2 (1999) 157–175

DOI: 10.2140/gtm.1999.2.157

arXiv: math.GT/9902122


Let p be an odd prime and r be relatively prime to p. Let G be a finite p–group. Suppose an oriented 3–manifold \tilde M has a free G–action with orbit space M. We consider certain Witten–Reshetikhin–Turaev SU(2) invariants wr(M) in Z[1/2r,e2πi/8r]. We will show that wr(~M)≡κ3 def(~M→M)(wr(M))|G| (mod p). Here κ=e2πi(r-2)/8r, def denotes the signature defect, and |G| is the number of elements in G. We also give a version of this result if M and ~M contain framed links or colored fat graphs. We give similar formulas for non-free actions which hold for a specified finite set of values for r.


p–group action, lens space, quantum invariant, Turaev–Viro invariant, branched cover, Jones polynomial, Arf invariant

Mathematical Subject Classification

Primary: 57M10

Secondary: 57M12


Received: 23 February 1999
Revised: 26 May 1999
Published: 18 November 1999

Patrick M Gilmer
Department of Mathematics
Louisiana State University
Baton Rouge LA 70803