2016/08/30 by Tadayuki Watanabe, Watanabe, Tadayuki
Computer Science · Mathematics · #57M27 #57R57 #58D29 #58E05 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1608.08462
openalex publication_date 2016/08/30 · openalex created_date 2016/09/16 · openalex updated_date 2026/07/28
Garoufalidis and Levine defined a filtration for 3-manifolds equipped with some degree 1 map (ℤπ-homology equivalence) to a fixed 3-manifold N and showed that there is a natural surjection from a space of π=π1N-decorated graphs to the graded quotient of the filtration over ℤ[(1)/(2)]. In this paper, we show that in the case of N=T3 the surjection of Garoufalidis--Levine is actually an isomorphism over ℚ. For the proof, we construct a perturbative invariant by applying Fukaya's Morse homotopy theoretic construction to a local system of the quotient field of ℚπ. The first invariant is an extension of the Casson invariant to ℤπ-homology equivalences to the 3-torus. The results of this paper suggest that there is a highly nontrivial equivariant quantum invariants for 3-manifolds with b1=3. We also discuss some generalizations of the perturbative invariant for other target spaces N.