Morse theory and Lescop's equivariant propagator for 3-manifolds with b₁=1 fibered over S¹

2014/03/31 by Tadayuki Watanabe, Watanabe, Tadayuki · 1 citation
Mathematics · #57M27 #57R57 #58D29 #58E05 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #math.DS #math.GT #msc:57M27 #msc:57R57 #msc:58D29 #msc:58E05

paper · pdf · doi:10.48550/arxiv.1403.8030

43 pages, 10 figures, v4, revised Theorem 1.2, Proposition 1.3 (supplement about orientability of fiberwise Morse function), Theorem 4.6 (AL-cycle must be replaced with the irreducible one), and made other few corrections

arxiv created 2014/09/02 · arxiv updated 2014/09/03

Abstract

For a 3-manifold M with b₁(M)=1 fibered over S¹ and the fiberwise gradient ξ of a fiberwise Morse function on M, we introduce the notion of amidakuji-like path (AL-path) on M. An AL-path is a piecewise smooth path on M consisting of edges each of which is either a part of a critical locus of ξ or a flow line of -ξ. Counting closed AL-paths with signs gives the Lefschetz zeta function of M. The "moduli space" of AL-paths on M gives explicitly Lescop's equivariant propagator, which can be used to define ℤ-equivariant version of Chern--Simons perturbation theory for M.

Cited by

• An invariant of fiberwise Morse functions on surface bundle over S¹ by counting graphs

Related

• An invariant of fiberwise Morse functions on surface bundle over S¹ by counting graphs
• Morse theory in path space
• On the stratification of a compact 3-manifold by the trajectory spaces of a Morse-Smale flow
• Morse theory in the 1990's
• The Morse Complex for a Morse Function on a Manifold with Corners