arXiv subjects

Toshihiro Yamaguchi

Publications and source records attributed to Toshihiro Yamaguchi.

At least 19 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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arXiv2026

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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arXiv2026

The group of homotopy self-equivalences is a Lax functor

The group $\E(X)$ of homotopy self-equivalences of a topological space $X$ is a well-known group in homotopy theory and has been studied by many people since it was first introduced in the late 1950s. $\E$ is not a functor in the usual sense. In this paper we show that $\E$ is a Lax functor from the category $\mathscr Top$ of topological spaces to a strict $2$-category $\op{Corr}_{\mathscr Gr}$ of \emph{correspondences} of groups.

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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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arXiv2020

Poincaré polynomials of a map and a relative Hilali conjecture

In this paper we introduce homological and homotopical Poincaré polynomials $P_f(t)$ and $P^π_f(t)$ of a continuous map $f:X \to Y$ such that if $f:X \to Y$ is a constant map, or more generally, if $Y$ is contractible, then these Poincaré polynomials are respectively equal to the usual homological and homotopical Poincaré polynomials $P_X(t)$ and $P^π_X(t)$ of the source space $X$. Our relative Hilali conjecture $P^π_f(1) \leqq P_f(1)$ is a map version of the the well-known Hilali conjecture $P^π_X(1) \leqq P_X(1)$ of a rationally elliptic space X. In this paper we show that under the condition that $H_i(f;\mathbb Q):H_i(X;\mathbb Q) \to H_i(Y;\mathbb Q)$ is not injective for some $i>0$, the relative Hilali conjecture of product of maps holds, namely, there exists a positive integer $n_0$ such that for $\forall n \geqq n_0$ the \emph{strict inequality $P^π_{f^n}(1) < P_{f^n}(1)$} holds, where $f^n:X^n \to Y^n$. In the final section we pose a question whether a "Hilali"-type inequality $HP^π_X(r_X) \leqq P_X(r_X)$ holds for a rationally hyperbolic space $X$, provided the the homotopical Hilbert--Poincare series $HP^π_X(r_X)$ converges at the radius $r_X$ of convergence.

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arXiv2019

Self-closeness numbers of finite cell complexes

We reformulate the inequalities among self-closeness numbers of spaces in cofibrations making use of homology dimension and show that the self-closeness number of a space is less than or equal to the homology dimension of the space. Then we prove a relation of self-closeness numbers and the connectivity for manifolds satisfying Poincaré duality. On the other hand we determine the self-closeness numbers of the real projective spaces, lens spaces and a cell complex defined by Mimura and Toda. Moreover, making use of the models of Sullivan and Quillen, we show several properties of self-closeness number for finite cell complexes, and rational examples are udied to obtain some precise results. Finally, we prove relations among self-closeness numbers defined by homotopy groups, homology groups and cohomology groups.

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arXiv2018

Poset-stratified space structures of homotopy sets

A poset-stratified space is a pair $(S, S \xrightarrow πP)$ of a topological space $S$ and a continuous map $π: S \to P$ with a poset $P$ considered as a topological space with its associated Alexandroff topology. In this paper we show that one can impose such a poset-stratified space structure on the homotopy set $[X, Y]$ of homotopy classes of continuous maps by considering a canonical but non-trivial order (preorder) on it, namely we can capture the homotopy set $[X, Y]$ as an object of the category of poset-stratified spaces. The order we consider is related to the notion of \emph{dependence of maps} (by Karol Borsuk). Furthermore via homology and cohomology the homotopy set $[X,Y]$ can have other poset-stratified space structures. In the cohomology case, we get some results which are equivalent to the notion of \emph{dependence of cohomology classes} (by René Thom) and we can show that the set of isomorphism classes of complex vector bundles can be captured as a poset-stratified space via the poset of the subrings consisting of all the characteristic classes. We also show that some invariants such as Gottlieb groups and Lusternik--Schnirelmann category of a map give poset-stratified space structures to the homotopy set $[X,Y]$

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arXiv2018

An induced map between rationalized classifying spaces for fibrations

Let $B{ aut}_1X$ be the Dold-Lashof classifying space of orientable fibrations with fiber $X$. For a rationally weakly trivial map $f:X\to Y$, our strictly induced map $a_f: (Baut_1X)_0\to (Baut_1Y)_0$ induces a natural map from a $X_0$-fibration to a $Y_0$-fibration. It is given by a map between the differential graded Lie algebras of derivations of Sullivan models. We note some conditions that the map $a_f$ admits a section and note some relations with the Halperin conjecture. Furthermore we give the obstruction class for a lifting of a classifying map $h: B\to (Baut_1Y)_0$ and apply it for liftings of $G$-actions on $Y$ for a compact connected Lie group $G$ as the case of $B=BG$ and evaluating of rational toral ranks as $r_0(Y)\leq r_0(X)$.

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arXiv2016

Certain maps preserving self-homotopy equivalences

Let $\mathcal{E}(X)$ be the group of homotopy classes of self homotopy equivalences for a connected CW complex $X$. We observe two classes of maps $\mathcal{E}$-maps and co-$\mathcal{E}$-maps. They are defined as the maps $X\to Y$ that induce the homomorphisms $\mathcal{E}(X)\to \mathcal{E}( Y)$ and $\mathcal{E}(Y)\to \mathcal{E}(X)$, respectively. We give some rationalized examples related to spheres, Lie groups and homogeneous spaces by using Sullivan models. Furthermore, we introduce an $\mathcal{E}$-equivalence relation between rationalized spaces $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$ as a geometric realization of an isomorphism $\mathcal{E}(X_{\mathbb{Q}})\cong \mathcal{E}(Y_{\mathbb{Q}})$.

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arXiv2015

Sullivan minimal models of classifying spaces for non-formal spaces of small rank

We consider certain rational homotopical conditions of simly connected CW complex $X$ such that the rational cohomology of the classifying space $Baut_1X$ for fibrations with two-stage fibre $X$ is (not) free. First, we consider when is $Baut_1X$ a rational factor of $Baut_1(X\times S^n)$ for an odd-integer $n$ and observe for a non-formal elliptic space $X$ of rank 3. Second, we compute the Sullivan minimal models of $Baut_1X$ when $X$ are certain non-formal pure spaces of rank 5.

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arXiv2013

Rational toral rank of a map

Let $X$ and $Y$ be simply connected CW complexes with finite rational cohomologies. The rational toral rank $r_0(X)$ of a space $X$ is the largest integer $r$ such that the torus $T^r$ can act continuously on a CW-complex in the rational homotopy type of $X$ with all its isotropy subgroups finite \cite{H}. As a rational homotopical condition to be a toral map preserving almost free toral actions for a map $f:X\to Y$, we define the rational toral rank $r_0(f)$ of $f$, which is a natural invariant with $r_0(id_X)=r_0(X)$ for the identity map $id_X$ of $X$. We will see some properties of it by Sullivan models, which is a free commutative differential graded algebra over $\Q$ \cite{FHT}.

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arXiv2013

C-symplectic poset structure on a simply connected space

For a field $\K$ of characteristic zero, we introduce a cohomologically symplectic poset structure ${\mathcal P}_{\K}(X)$ on a simply connected space $X$ from the viewpoint of $\K$-homotopy theory. It is given by the poset of inclusions of subgroups preserving c-symplectic structures in the group ${\mathcal E}(X_{\K})$ of $\K$-homotopy classes of $\K$-homotopy self-equivalences of $X$, which is defined by the $\K$-Sullivan model of $X$. We observe that the height of the Hasse diagram of ${\mathcal P}_{\K}(X)$ added by 1, denoted by c-s-${\rm depth}_{\K}(X)$, is finite and often depends on the field $\K$. In this paper, we will give some examples of ${\mathcal P}_{\K}(X)$.

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arXiv2013

A rational realization problem in Gottlieb group

We define the fibre-restricted Gottlieb group with respect to a fibration $ξ:X\to E\to Y$ in CW complexes. It is a subgroup of the Gottlieb group of $X$. When $X$ and $E$ are finite simply connected, its rationalized model is given by the arguments of derivations of Sullivan models based on Félix, Lupton and Smith \cite{FLS}. We consider the realization problem of groups in a Gottlieb group as fibre-restricted Gottlieb groups in rational homotoy theory. Especially we define an invariant named as (Gottlieb) depth of $X$ over $Y$. In particular, when $Y=BS^1$, it is related to the rational toral rank of $X$.

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arXiv2012

Pre-c-symplectic condition for the product of odd-spheres

We say that a simply connected space $X$ is pre-c-symplectic if it is the fibre of a rational fibration $X\to Y\to \C P^{\infty}$ where $Y$ is cohomologically symplectic in the sense that there is a degree 2 cohomology class which cups to a top class. It is a rational homotopical property but not a cohomological one. By using Sullivan's minimal models, we give the necessary and sufficient condition that the product of odd-spheres $X=S^{k_1}\times ... \times S^{k_n}$ is pre-c-symplectic and see some related topics. Also we give a charactarization of the Hasse diagram of rational toral ranks for a space $X$ as a necessary condition to be pre-c-symplectic and see some examples in the cases of finite-oddly generated rational homotopy groups.

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arXiv2012

Examples of rational toral rank complex

In "A Hosse diagram for rational toral tanks," we see a CW complex ${\mathcal T}(X)$, which gives a rational homotopical classification of almost free toral actions on spaces in the rational homotopy type of $X$ associated with rational toral ranks and also presents certain relations in them. We call it the {\it rational toral rank complex} of $X$. It represents a variety of toral actions. In this note, we will give effective 2-dimensional examples of it when $X$ is a finite product of odd spheres. This is a combinatorial approach in rational homotopy theory.

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arXiv2010

A Hasse diagram for rational toral ranks

Let $X$ be a simply connected CW complex with finite rational cohomology. For the finite quotient set of rationalized orbit spaces of $X$ obtained by almost free toral actions, ${\mathcal T}_0(X)=\{[Y_i] \}$, induced by an equivalence relation based on rational toral ranks, we order as $[Y_i]<[Y_j]$ if there is a rationalized Borel fibration $Y_i\to Y_j\to BT^n_{\Q}$ for some $n>0$. It presents a variation of almost free toral actions on $X$. We consider about the Hasse diagram ${\mathcal H}(X)$ of the poset ${\mathcal T}_0(X)$, which makes a based graph $G{\mathcal H}(X)$, with some examples. Finally we will try to regard $G{\mathcal H}(X)$ as the 1-skeleton of a finite CW complex ${\mathcal T}(X)$ with base point $X_{\Q}$.

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arXiv2010

A relaxed evaluation subgroup

Let $f:X\to Y$ be a pointed map between connected CW-complexes. As a generalization of the evaluation subgroup $G_*(Y,X;f)$, we will define the {\it relaxed evaluation subgroup} ${\mathcal G}_*(Y,X;f)$ in the homotopy group $π_*(Y)$ of $Y$, which is identified with ${\rm Im} π_*(\tilde{ev})$ for the evaluation map $\tilde{ev} :map(X,Y;f)\times X\to Y$ given by $\tilde{ev} (h,x)=h(x)$. Especially we see by using Sullivan model in rational homotopy theory for the rationalized map $f_{\Q}$ that ${\mathcal G}_*(Y_{\Q},X_{\Q};f_{\Q})=π_*(Y)\otimes \Q$ if the map $f$ induces an injection of rational homotopy groups. Also we compare it with more relaxed subgroups by several rationalized examples.

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