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Shun Wakatsuki

Publications and source records attributed to Shun Wakatsuki.

12 recordsLinked to original sources
arXiv2026

A distance between maps via interleavings of relative Sullivan algebras

In this article, we consider extended tame persistence commutative differential graded algebras (CDGAs) associated with relative Sullivan algebras. In particular, if the relative Sullivan algebra is a model for a map between spaces, then the persistence CDGA is isomorphic to the persistence object obtained by a Postnikov tower for the map with the polynomial de Rham functor in the homotopy category of extended tame persistence CDGAs. Moreover, the interleaving distance in the homotopy category (IHC) in the sense of Lanari and Scoccola enables us to introduce a pseudodistance on the homotopy set of maps via the persistence CDGA models for maps. In contrast to persistence cochain complexes, the IHC of persistence CDGAs does not coincide with the cohomology interleaving distance in general. Due to the reason, we also discuss formalities of a persistence CDGA with interleavings. Computational examples of the pseudodistances between maps are showcased.

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arXiv2025

The equalities of interleaving distances and cohomology interleavings of spaces over $BS^1$

The cohomology interleaving distance (CohID) is defined and considered in the category of persistent differential graded modules over a field. As a consequence, we show that, in the category, the distance coincides with the homotopy commutative interleaving distance, the homotopy interleaving distance originally due to Blumberg and Lesnick, and the interleaving distance in the homotopy category in the sense of Lanari and Scoccola. Moreover, we apply the CohID to spaces over the classifying space $BS^1$ of the circle group via the singular cochain functor. Then, upper and lower bounds of the CohID are investigated with the cup-lengths of spaces over $BS^1$. As a computational example, we explicitly determine the CohID for complex projective spaces by utilizing the bottleneck distance of barcodes associated with the cohomology of the spaces.

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arXiv2024

Dominance complexes, neighborhood complexes and combinatorial Alexander duals

We show that the dominance complex $\mathcal{D}(G)$ of a graph $G$ coincides with the combinatorial Alexander dual of the neighborhood complex $\mathcal{N}(\overline{G})$ of the complement of $G$. Using this, we obtain a relation between the chromatic number $χ(G)$ of $G$ and the homology group of $\mathcal{D}(G)$. We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest a new method for computing the homology groups of the dominance complexes, using independence complexes of simple graphs. We show that several known computations of homology groups of dominance complexes can be reduced to known computations of independence complexes. Finally, we determine the homology group of $\mathcal{D}(P_n \times P_3)$ by determining the homotopy types of the independence complex of $P_n \times P_3 \times P_2$.

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arXiv2024

Minimal projective resolution and magnitude homology of geodetic metric spaces

Asao-Ivanov showed that magnitude homology is a Tor functor, hence we can compute it by giving a projective resolution of a certain module. In this article, we compute magnitude homology by constructing a minimal projective resolution. As a consequence, we determine magnitude homology of geodetic metric spaces. We show that it is a free $\mathbb Z$-module, and give a recursive algorithm for constructing all cycles. As a corollary, we show that a finite geodetic metric space is diagonal if and only if it contains no 4-cuts. Moreover, we give explicit computations for cycle graphs, Petersen graph, Hoffman-Singleton graph, and a missing Moore graph. It includes another approach to the computation for cycle graphs, which has been studied by Hepworth--Willerton and Gu.

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arXiv2024

Cartan calculi on the free loop spaces

A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is investigated. In a general setting, the stage is formulated with operators obtained by the André-Quillen cohomology of a commutative differential graded algebra $A$ on the Hochschild homology of $A$ in terms of the homotopy Cartan calculus in the sense of Fiorenza and Kowalzig. Moreover, the Cartan calculus is interpreted geometrically with maps from the rational homotopy group of the monoid of self-homotopy equivalences on a space $M$ to the derivation ring on the loop cohomology of $M$. We also give a geometric description to Sullivan's isomorphism, which relates the geometric Cartan calculus to the algebraic one, via the $Γ_1$ map due to Félix and Thomas.

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arXiv2023

A reduction of the string bracket to the loop product

The negative cyclic homology for a differential graded algebra over the rational field has a quotient of the Hochschild homology as a direct summand if the $S$-action is trivial. With this fact, we show that the string bracket in the sense of Chas and Sullivan is reduced to the loop product followed by the BV operator on the loop homology provided the given manifold is BV exact. The reduction is indeed derived from the equivalence between the BV exactness and the triviality of the $S$-action. Moreover, it is proved that a Lie bracket on the loop cohomology of the classifying space of a connected compact Lie group possesses the same reduction. By using these results, we consider the non-triviality of string brackets. Another highlight is that a simply-connected space with positive weights is BV exact. Furthermore, the higher BV exactness is also discussed featuring the cobar-type Eilenberg-Moore spectral sequence.

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arXiv2019

New construction of the brane coproduct and vanishing of cup products on sphere spaces

Using the loop coproduct, Menichi proved that the cup product with the orientation class vanishes for a closed connected oriented manifold with non-trivial Euler characteristic. We generalize this to the sphere spaces, i.e. the mapping spaces from spheres, using two generalizations of the loop coproduct to sphere spaces. One is constructed in this paper and the other in a previous paper of the author.

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arXiv2018

Coproducts in brane topology

We extend the loop product and the loop coproduct to the mapping space from the $k$-dimensional sphere, or more generally from any $k$-manifold, to a $k$-connected space with finite dimensional rational homotopy group, $k\geq 1$. The key to extending the loop coproduct is the fact that the embedding $M\rightarrow M^{S^{k-1}}$ is of "finite codimension" in a sense of Gorenstein spaces. Moreover, we prove the associativity, commutativity, and Frobenius compatibility of them.

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arXiv2016

Description and triviality of the loop products and coproducts for rational Gorenstein spaces

Félix and Thomas extended the loop products and coproducts to simply-connected Gorenstein spaces. We explicitly describe these operations with rational coefficients in terms of Sullivan models. Moreover, by this description, we prove some results on triviality of these operations. They include a variant of the result of Tamanoi, and generalizations of that of Félix and Thomas and that of Naito.

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