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Takefumi Nosaka

Publications and source records attributed to Takefumi Nosaka.

At least 19 recordsLinked to original sources
arXiv2026

Gröbner Bases for Alexander Fitting Ideals of Links

Multivariable Alexander theory produces Fitting ideals without preferred generators. After fixing a coefficient field and a monomial order, we represent these ideals by reduced Gröbner bases of their polynomial contractions, obtaining Alexander--Gröbner invariants of links. Over \(\mathbb Q\), for the maximal free-abelian coefficient system of a connected compact oriented \(3\)-manifold whose nonempty boundary is a disjoint union of tori, we deduce determinant-divisor reciprocity from Blanchfield duality. We compute these invariants for torus links and low-crossing links, and we determine \(\AG_2\) for a two-component pretzel family.

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arXiv2026

Chern--Simons-type $3$-cocycles and $\mathbb{Q}/\mathbb{Z}$-torsion in the third homology of discrete diffeomorphism groups

We prove that the third integral group homology of both the volume-preserving diffeomorphism group and the strict contactomorphism group of the standard \(3\)-sphere contains \((\mathbb Q/\mathbb Z)^2\). The proof constructs two independent locally smooth Chern--Simons-type \(3\)-classes from the volume form and the standard framing of \(S^3\). A torsion-controlled local integration method and comparison with the primitive Cheeger--Chern--Simons class also detect \(\mathbb Q/\mathbb Z\) for several spherical space forms and symplectic projective manifolds.

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arXiv2026

Type Annihilation for Classifying Maps of Rack Spaces

We study the classifying map $c\colon BX\to K(\As(X),1)$ of a rack $X$ of finite type. Let $t = \Type(X)$. We prove that $t c_{n*}=0$ for every $n\ge2$ when $X$ is connected, and that $t^{n-1}c_{n*}=0$ on the torsion subgroup $\Tor H_n^\mathbb{R}(X)$ without any connectedness assumption. For a finite rack, under our sign conventions, the rationalized classifying map in degree $n$ is given by $(-1)^n$ times the canonical projection from the $n$-fold tensor power of the orbit module to its $n$-th exterior power. For an arbitrary rack of finite type, we determine $H_2^{\mathrm{gr}}(\As(X);\mathbb{Z}[1/t])$. We also derive low-dimensional applications to symplectic and Alexander structures.

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arXiv2024

Nonacyclic Reidemeister torsions of manifolds of odd dimension

Given an oriented closed manifold $M$ of odd dimension and a unitary representation $ρ: π_1(M) \ra \GL_n(\F)$, we define a Reidemeister torsion, even if the cohomology associated with $ρ$ is not acyclic. As corollaries, we introduce some topological invariants of $M$, which include the nonacyclic extensions of abelian torsions and the Alexander polynomials of links. Further, we propose a volume form of the $\SU(n)$-character varieties of $M$. Moreover, we compute the Reidemeister torsions of some representations of 3-manifolds and compare the works of Farber--Turaev.

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arXiv2024

The Dijkgraaf-Witten invariants from second Chern classes

Given a 3-cocycle $ψ$ in the cohomology of a finite group $G$, we can define the Dijkgraaf-Witten invariant of closed 3-manifolds. In this paper, we focus on the case where $ψ$ is a 3-cocycle canonically obtained from the second Chern class of a representation of $G$, and show some procedures for computing the invariant, and clarify its topological interpretation under some conditions.

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arXiv2023

A solvable extended logarithm of the Johnson homomorphism

We suggest an extension of a certain logarithm of the total Johnson map in terms of solvable Lie groups. Here, the domain of the map is extended to a subset consisting of exponential solvable elements in the mapping class group of a surface.

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arXiv2023

Goldman-type Lie algebras from knots

We define Lie algebras from a class of knots in a homology 3-sphere. Since the definitions in terms of group homology are analogous to Goldman Lie algebra \cite{Gold}, we discuss relations among these Lie algebras.

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arXiv2023

Skew-rack cocycle invariants of closed 3-manifolds

We establish a new approach to obtain 3-manifold invariants via Dehn surgery. For this, we introduce skew-racks with good involution and Property FR, and define cocycle invariants as 3-manifold invariants. We also define some link invariants in the 3-sphere which are invariant up to link-homotopic.

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arXiv2022

Reciprocity of the Chern-Simons invariants of 3-manifolds

We pose reciprocity conjectures of the Chern-Simons invariants of 3-manifolds, and discuss some supporting evidence on the conjectures. Especially, we show that the conjectures hold if a Galois descent of a $K_3$-group is satisfied.

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arXiv2022

Cellular chain complexes of universal covers of some 3-manifolds

For a closed 3-manifold $M$ in a certain class, we give a presentation of the cellular chain complex of the universal cover of $M$. The class includes all surface bundles, some surgeries of knots in $S^3$, some cyclic branched cover of $S^3$, and some Seifert manifolds. In application, we establish a formula for calculating the linking form of a cyclic branched cover of $S^3$, and develop procedures of computing some Dijkgraaf-Witten invariants.

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arXiv2021

Fox pairings of Poincaré duality groups

This paper develops the study of Fox pairings of a group $G$ from the viewpoint of group cohomology. We compute some cohomology groups of Fox pairings of $G$, where $G$ admits a Poincaré duality group pair. We also suggest fundamental Fox pairings and higher Fox pairings.

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arXiv2020

Some comparisons of Blanchfield pairings and cohomology pairings of knots

We study some comparison between a bilinear cohomology pairing in local coefficients and the Blanchfield pairing of a knot. We show that the former pairing is an $S$-equivalent invariant, and give a criterion to a relation between the two pairings. We also observe that the pairings of some knots are equivalent, and that the pairings of other knots are not equivalent.

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arXiv2020

Twisted cohomology pairings of knots II; to classical invariants

We show that bilinear cup products with local coefficients of closed 3-manifolds recover some twisted pairings of infinite covers and the Casson-Gordon local signatures. As a result, we further give diagrammatic computations of the pairings and signatures.

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arXiv2020

Twisted Alexander invariants of knot group representations

Given a homomorphism from a knot group to a fixed group, we introduce an element of a $K_1$-group, which is a generalization of (twisted) Alexander polynomials. We compare this $K_1$-class with other Alexander polynomials. In terms of semi-local rings, we compute the $K_1$-classes of some knots and show their non-triviality. We also introduce metabelian Alexander polynomials.

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arXiv2020

Twisted Alexander invariants of knot group representations II; computation and duality

Given a homomorphism from a link group to a group, we introduce a $K_1$-class in another way, which is a generalization of the 1-variable Alexander polynomial. We compare the $K_1$-class with $K_1$-classes in \cite{Nos} and with Reidemeister torsions. As a corollary, we show a relation to Reidemeister torsions of finite cyclic covering spaces, and show reciprocity in some senses.

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arXiv2019

Milnor invariants via unipotent Magnus embeddings

We reconfigure the Milnor invariant of links in terms of central group extensions and unipotent Magnus embeddings. We also develop a diagrammatic computation of the invariant and compute the first non-vanishing invariants of the Milnor link and of several other links. Moreover, we refine the original Milnor invariants of higher degree.

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