arXiv subjects

Shoji Yokura

Publications and source records attributed to Shoji Yokura.

At least 19 recordsLinked to original sources

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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arXiv2025

A co-operational bivariant theory derived from cohomology operations

A co-operational bivariant theory is a ``dual" version of Fulton--MacPherson's operational bivaiant theory. For a given contravariant functor we define a generalized cohomology operation for continuous maps having sections, using cohomology operations. This generalized cohomology operation is related to Quillen's Steenrod power operation. Using this generalized cohomology operation, we define \emph{a co-operational bivariant theory derived from cohomology operations}, which is a subtheory of the co-operational bivariant theory associated to the contravariant functor.

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arXiv2025

On stratifications and poset-stratified spaces

A stratified space is a topological space equipped with a \emph{stratification}, which is a decomposition or partition of the topological space satisfying certain extra conditions. More recently, the notion of poset-stratified space, i.e., topological space endowed with a continuous map to a poset with its Alexandrov topology, has been popularized. Both notions of stratified spaces are ubiquitous in mathematics, ranging from investigations of singular structures in algebraic geometry to extensions of the homotopy hypothesis in higher category theory. In this article we study the precise mathematical relation between these different approaches to stratified spaces.

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arXiv2024

Hilali conjecture and complex algebraic varieties

A simply connected topological space is called \emph{rationally elliptic} if the rank of its total homotopy group and its total (co)homology group are both finite. A well-known Hilali conjecture claims that for a rationally elliptic space its homotopy rank \emph{does not exceed} its (co)homology rank. In this paper, after recalling some well-known fundamental properties of a rationally elliptic space and giving some important examples of rationally elliptic spaces and rationally elliptic singular complex algebraic varieties for which the Hilali conjecture holds, we give some revised formulas and some conjectures. We also discuss some topics such as mixd Hodge polynomials defined via mixed Hodge structures on cohomology group and the dual of the homotopy group, related to the ``Hilali conjecture \emph{modulo product}", which is an inequality between the usual homological Poincaré polynomial and the homotopical Poincaré polynomial.

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arXiv2024

A bi-variant algebraic cobordism via correspondences

A bi-variant theory $\mathbb B(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton--MacPherson's bivariant theory $\mathbb B(X \xrightarrow f Y)$ defined for a morphism $f:X \to Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism $Ω^{*,\sharp}(X, Y)$ such that $Ω^{*,\sharp}(X, pt)$ is isomorphic to Lee--Pandharipande's algebraic cobordism of vector bundles $Ω_{-*,\sharp}(X)$. In particular, $Ω^{*}(X, pt)=Ω^{*, 0}(X, pt)$ is isomorphic to Levine--Morel's algebraic cobordism $Ω_{-*}(X)$. Namely, $Ω^{*,\sharp}(X, Y)$ is \emph{a bi-variant vesion} of Lee--Pandharipande's algebraic cobordism of bundles $Ω_{*,\sharp}(X)$.

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arXiv2024

Co-operational bivariant theory

For a covariant functor W. Fulton and R. MacPherson defined \emph{an operational bivariant theory} associated to this covariant functor. In this paper we will show that given a contravariant functor one can similarly construct a ``dual" version of an operational bivariant theory, which we call a \emph{co-operational} bivariant theory. If a given contravariant functor is the usual cohomology theory, then our co-operational bivariant group for the identity map consists of what are usually called ``cohomology operations". In this sense, our co-operational bivariant theory consists of \emph{``generalized"} cohomology operations.

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arXiv2021

Self-products of rationally elliptic spaces and inequalities between the ranks of homotopy and homology groups

We give a survey on recent results on inequalities between the ranks of homotopy and cohomology groups (resp., graded components of mixed Hodge structures on these groups) of rationally elliptic spaces (resp., quasi-projective varieties which are rationally elliptic). We also discuss a refinement of these results describing a new invariant of rationally elliptic spaces allowing to compare the ranks of homotopy and homology groups. This invariant is a specialization of an invariant $r\left ( P(t),Q(t);\varepsilon \right )$ of a pair $\left (P(t),Q(t) \right )$ of polynomials with non-negative integer coefficients, describing the range of variable $r$ such that $rP(t)<Q(t)^r$ for all $t \ge \varepsilon$. This range is related to the classical Lambert W-function $W(z)$.

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arXiv2021

Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties

We discuss inequalities between the values of \emph{homotopical and cohomological Poincaré polynomials} of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective varieties, we prove inequalities between the values of generating functions for the ranks of the graded pieces of the weight and Hodge filtrations of the canonical mixed Hodge structures on homotopy and cohomology groups. Several examples of such mixed Hodge polynomials and related inequalities for rationally elliptic quasi-projective algebraic varieties are presented. One of the consequences is that the homotopical (resp. cohomological) mixed Hodge polynomial of a rationally elliptic toric manifold is a sum (resp. a product) of polynomials of projective spaces. We introduce an invariant called \emph{stabilization threshold} $\frak{pp} (X;\varepsilon)$ for a simply connected rationally elliptic space $X$ and a positive real number $\varepsilon$, and we show that the Hilali conjecture implies that $\frak{pp} (X;1) \le 3$.

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arXiv2020

Cobordism bicycles of vector bundles

The main ingredient of the algebraic cobordism of M. Levine and F. Morel is a cobordism cycle of the form $(M \xrightarrow {h} X; L_1, \cdots, L_r)$ with a proper map $h$ from a smooth variety $M$ and line bundles $L_i$'s over $M$. In this paper we consider a \emph{cobordism bicycle} of a finite set of line bundles $(X \xleftarrow p V \xrightarrow s Y; L_1, \cdots, L_r)$ with a proper map $p$ and a smooth map $s$ and line bundles $L_i$'s over $V$. We will show that the Grothendieck group $\mathscr Z^*(X, Y)$ of the abelian monoid of the isomorphism classes of cobordism bicycles of finite sets of line bundles satisfies properties similar to those of Fulton-MacPherson's bivariant theory and also that $\mathscr Z^*(X, Y)$ is a \emph{universal} one among such abelian groups, i.e., for any abelian group $\mathscr B^*(X, Y)$ satisfying the same properties there exists a unique Grothendieck transformation $γ: \mathscr Z^*(X,Y) \to \mathscr B^*(X,Y)$ preserving the unit.

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arXiv2020

Enriched categories of correspondences and characteristic classes of singular varieties

For the category $\mathscr V$ of complex algebraic varieties, the Grothendieck group of the commutative monoid of the isomorphism classes of correspondences $X \xleftarrow f M \xrightarrow g Y$ with proper morphism $f$ and smooth morphism $g$ (such a correspondence is called \emph{a proper-smooth correspondence}) gives rise to an enriched category $\mathscr Corr(\mathscr V)^+_{pro-sm}$ of proper-smooth correspondences. In this paper we extend the well-known theories of characteristic classes of singular varieties such as Baum-Fulton-MacPherson's Riemann-Roch (abbr. BFM-RR) and MacPherson's Chern class transformation and so on to this enriched category $\mathscr Corr(\mathscr V)^+_{pro-sm}$. In order to deal with local complete intersection (abbr. $\ell.c.i.$) morphism instead of smooth morphism, in a similar manner we consider an enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$ of \emph{proper-$\ell.c.i.$} zigzags and extend BFM-RR to this enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$. We also consider an enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ of proper-smooth correspondences $(X \xleftarrow f M \xrightarrow g Y; E)$ equipped with complex vector bundle $E$ on $M$ (such a correspondence is called \emph{a cobordism bicycle of vector bundle}) and we extend BFM-RR to this enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ as well.

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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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arXiv2020

A remark on periods of periodic sequences modulo $m$

Let $\{G_n\}$ be a periodic sequence of integers modulo $m$ and let $\{SG_n\}$ be the partial sum sequence defined by $SG_n:= \sum_{k=0}^nG_k $ (mod $m$). We give a formula for the period of $\{SG_n\}$. We also show that for a generalized Fibonacci sequence $F(a,b)_n$ such that $F(a,b)_0=a$ and $F(a,b)_1=b$, we have $$S^i F(a,b)_n = S^{i-1}F(a,b)_{n+2}-{n+i \choose i-2}a-{n+i \choose i-1} b$$ where $S^i F(a,b)_n $ is the i-th partial sum sequence successively defined by $S^i F(a,b)_n := \sum_{k=0}^n S^{i-1}F(a,b)_k$. This is a generalized version of the well-known formula $$\sum_{k=0}^n F_k = F_{n+2} -1$$ of the Fibonacci sequence $F_n$.

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arXiv2020

Bivariant algebraic cobordism with bundles

The purpose of this paper is to study an extended version of bivariant derived algebraic cobordism where the cycles carry a vector bundle on the source as additional data. We show that, over a field of characteristic 0, this extends the analogous homological theory of Lee and Pandharipande constructed earlier. We then proceed to study in detail the restricted theory where only rank 1 vector bundles are allowed, and prove a weak version of projective bundle formula for bivariant cobordism. Since the proof of this theorem works very generally, we introduce precobordism theories over arbitrary Noetherian rings of finite Krull dimension as a reasonable class of theories where the proof can be carried out, and prove some of their basic properties. These results can be considered as the first steps towards a Levine-Morel style algebraic cobordism over a base ring that is not a field of characteristic 0.

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arXiv2020

Local comparisons of homological and homotopical mixed Hodge polynomials

For a simply connected complex algebraic variey $X$, by the mixed Hodge structures $(W_{\bullet}, F^{\bullet})$ and $(\tilde W_{\bullet}, \tilde F^{\bullet})$ of the homology group $H_{*}(X;\mathbb Q)$ and the homotopy groups $π_{*}(X)\otimes \mathbb Q$ respectively, we have the following mixed Hodge polynomials $$MH_X(t,u,v):= \sum_{k,p,q} \operatorname{dim} \Bigl ( Gr_{F^{\bullet}}^{p} Gr^{W_{\bullet}}_{p+q} H_k (X;\mathbb C) \Bigr) t^{k} u^{-p} v^{-q},$$ $$\quad \, \, MH^π_X(t,u,v):= \sum_{k,p,q} \operatorname{dim} \Bigl (Gr_{\tilde F^{\bullet}}^{p} Gr^{\tilde W_{\bullet}}_{p+q} (π_k(X) \otimes \mathbb C) \Bigr ) t^ku^{-p} v^{-q},$$ which are respectively called \emph{the homological mixed Hodge polynomial} and \emph{the homotopical mixed Hodge polynomial}. In this paper we discuss some inequalities concerning these two mixed Hodge polynomials.

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arXiv2019

The Hilali conjecture on product of spaces

The Hilali conjecture claims that a simply connected rationally elliptic space $X$ satisfies the inequality $\operatorname{dim} (π_*(X)\otimes \mathbb Q ) \leqq \operatorname{dim} H_*(X;\mathbb Q )$. In this paper we show that for any such space $X$ there exists a positive integer $n_0$ such that for any $n \geqq n_0$ the \emph{strict inequality} $\operatorname{dim} (π_*(X^n)\otimes \mathbb Q ) < \operatorname{dim} H_*(X^n;\mathbb Q )$ holds, where $X^{n}$ is the product of $n$ copies of $X$.

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arXiv2019

Decomposition spaces and poset-stratified spaces

In 1920s R. L. Moore introduced \emph{upper semicontinuous} and \emph{lower semicontinuous} decompositions in studying decomposition spaces. Upper semicontinuous decompositions were studied very well by himself and later by R.H. Bing in 1950s. In this paper we consider lower semicontinuous decompositions $\mathcal D$ of a topological space $X$ such that the decomposition spaces $X/\mathcal D$ are Alexandroff spaces. If the associated proset (preordered set) of the decomposition space $X/\mathcal D$ is a poset, then the decomposition map $π:X \to X/\mathcal D$ is \emph{a continuous map from the topological space $X$ to the poset $X/\mathcal D$ with the associated Alexandroff topology}, which is nowadays called \emph{a poset-stratified space}. As an application, we capture the face poset of a real hyperplane arrangement $\mathcal A$ of $\mathbb R^n$ as the associated poset of the decomposition space $\mathbb R^n/\mathcal D(\mathcal A)$ of the decomposition $\mathcal D(\mathcal A)$ determined by the arrangement $\mathcal A$. We also show that for any locally small category $\mathcal C$ the set $hom_{\mathcal C}(X,Y)$ of morphisms from $X$ to $Y$ can be considered as a poset-stratified space, and that for any objects $S, T$ (where $S$ plays as a source object and $T$ as a target object) there are a covariant functor $\frak {st}^S_*: \mathcal C \to \mathcal Strat$ and a contravariant functor $\frak {st}^*_T$ $\frak {st}^*_T: \mathcal C \to \mathcal Strat$ from $\mathcal C$ to the category $\mathcal Strat$ of poset-stratified spaces. We also make a remark about Yoneda's Lemmas as to poset-stratified space structures of $hom_{\mathcal C}(X,Y)$.

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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

Homological congruence formulae for characteristic classes of singular varieties

For a pair $(f, g)$ of morphisms $f:X \to Z$ and $g:Y \to Z$ of (possibly singular) complex algebraic varieties $X,Y,Z$, we present congruence formulae for the difference $f_*T_{y*}(X) -g_*T_{y*}(Y)$ of pushforwards of the corresponding motivic Hirzebruch classes $T_{y*}$. If we consider the special pair of a fiber bundle $F \hookrightarrow E \to B$ and the projection $pr_2:F \times B \to B$ as such a pair $(f,g)$, then we get a congruence formula for the difference $f_*T_{y*}(E) -χ_y(F)T_{y*}(B)$, which at degree level yields a congruence formula for $χ_y(E) -χ_y(F)χ_y(B)$, expressed in terms of the Euler--Poincarv'e characteristic, Todd genus and signature in the case when $F, E, B$ are non-singular and compact. We also extend the finer congruence identities of Rovi--Yokura to the singular complex projective situation, by using the corresponding intersection (co)homology invariants.

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