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We apply Lescops construction of $mathbb{Z}$-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of AL-paths given in arXiv:1403.8030. We obtain an invariant $hat{Z}_n$ of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over $S^1$, which can be considered as a higher loop analogue of the Lefschetz zeta function and whose construction will be applied to that of finite type invariants of knots in such a 3-manifold. We also give a combinatorial formula for Lescops equivariant invariant $mathscr{Q}$ for 3-manifolds with $H_1=mathbb{Z}$ fibered over $S^1$. Moreover, surgery formulas of $hat{Z}_n$ and $mathscr{Q}$ for alternating sums of surgeries are given. This gives another proof of Lescops surgery formula of $mathscr{Q}$ for special kind of 3-manifolds and surgeries, which is simple in the sense that the formula is obtained easily by counting certain graphs in a 3-manifold.
We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration ${mathscr{K}_n(M)}_{ngeq 0}$ of the vector space $mathscr{K}(M)$ spanned by isotopy classes of nullhomologous knots i
In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukayas Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functio
For a 3-manifold $M$ with $b_1(M)=1$ fibered over $S^1$ and the fiberwise gradient $xi$ 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
In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of pa
We give a generalization of Fukayas Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with