Garoufalidis-Levine's finite type invariants for $\mathbb{Z}π$-homology equivalences from 3-manifolds to the 3-torus
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arXiv
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| Format: | Preprint |
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2016
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| _version_ | 1866916187991965696 |
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| author | Watanabe, Tadayuki |
| author_facet | Watanabe, Tadayuki |
| contents | Garoufalidis and Levine defined a filtration for 3-manifolds equipped with some degree 1 map ($\mathbb{Z}π$-homology equivalence) to a fixed 3-manifold $N$ and showed that there is a natural surjection from a space of $π=π_1N$-decorated graphs to the graded quotient of the filtration over $\mathbb{Z}[\frac{1}{2}]$. In this paper, we show that in the case of $N=T^3$ the surjection of Garoufalidis--Levine is actually an isomorphism over $\mathbb{Q}$. 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 $\mathbb{Q}π$. The first invariant is an extension of the Casson invariant to $\mathbb{Z}π$-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 $b_1=3$. We also discuss some generalizations of the perturbative invariant for other target spaces $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1608_08462 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Garoufalidis-Levine's finite type invariants for $\mathbb{Z}π$-homology equivalences from 3-manifolds to the 3-torus Watanabe, Tadayuki Geometric Topology Algebraic Topology 57M27, 57R57, 58D29, 58E05 Garoufalidis and Levine defined a filtration for 3-manifolds equipped with some degree 1 map ($\mathbb{Z}π$-homology equivalence) to a fixed 3-manifold $N$ and showed that there is a natural surjection from a space of $π=π_1N$-decorated graphs to the graded quotient of the filtration over $\mathbb{Z}[\frac{1}{2}]$. In this paper, we show that in the case of $N=T^3$ the surjection of Garoufalidis--Levine is actually an isomorphism over $\mathbb{Q}$. 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 $\mathbb{Q}π$. The first invariant is an extension of the Casson invariant to $\mathbb{Z}π$-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 $b_1=3$. We also discuss some generalizations of the perturbative invariant for other target spaces $N$. |
| title | Garoufalidis-Levine's finite type invariants for $\mathbb{Z}π$-homology equivalences from 3-manifolds to the 3-torus |
| topic | Geometric Topology Algebraic Topology 57M27, 57R57, 58D29, 58E05 |
| url | https://arxiv.org/abs/1608.08462 |