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Norio Iwase

Publications and source records attributed to Norio Iwase.

At least 19 recordsLinked to original sources
arXiv2025

Associahedra, Multiplihedra and units in $A_{\infty}$ form

A higher associativity was introduced by Jim Stasheff in [Sta63] with higher coherence conditions and now becomes one of the most important structures on spaces and algebras. He also claims that the condition on unit can be weakened, using James retractile arguments [Jam60], while the proof given in [Sta63] for the equivalence of two definitions is not very clear for us. We had been puzzled for years, and decided to prove it in a different way by constructing an $A_{m}$-structure. To justify that our construction is natural, we bring our ideas into the theory of an internal precategory which is a weak version of Aguiar's internal category [Agu97]. Using that construction, we show the equivalence of two definitions under the `loop-like' condition. That condition is not necessary to manipulate higher forms using retractile arguments as is performed in [Sta63], but is necessary to construct an $A_{m}$-structure from the given $A_{m}$-form with {\em strict-unit} as is mentioned in Stasheff [Sta70].

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arXiv2025

A closed manifold is a fat CW complex

The main purpose of this paper is to introduce a new smooth version of a CW complex named a fat CW complex, and to show that it includes all closed manifolds, because existing smooth versions of CW complexes (e.g. [Iwa22]) do not have such property. We also verify that de Rham theorem holds for a fat CW complex and that a regular CW complex is reflexive in the sense of Y. Karshon, J. Watts and P. I-Zemmour. Further, any topological CW complex is topologically homotopy equivalent to a fat CW complex. So, a fat CW complex enjoys many nice properties.

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arXiv2023

Topological Complexity of $S^3/Q_8$ as fibrewise L-S category

In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $\operatorname{pr}_{1} : X \times X \to X$ with the diagonal map $Δ: X \to X \times X$ as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.

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arXiv2023

Smooth $A_{\infty}$ form on a diffeological loop space

To construct an $A_{\infty}$-form for a loop space in the category of diffeological spaces, we have two minor problems. Firstly, the concatenation of paths in the category of diffeological spaces needs a small technical trick (see P.~I-Zemmour \cite{MR3025051}), which apparently restricts the number of iterations of concatenations. Secondly, we do not know a natural smooth decomposition of an associahedron as a simplicial or a cubical complex. To resolve these difficulties, we introduce a notion of a $q$-cubic set which enjoys good properties on dimensions and representabilities, and show, using it, that the smooth loop space of a reflexive diffeological space is a h-unital smooth $A_{\infty}$-space. In appendix, we show an alternative solution by modifying the concatenation to be stable without assuming reflexivity for spaces nor stability for paths.

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arXiv2022

Lusternik-Schnirelmann theory to Topological Complexity from $A_{\infty}$-view point

We are trying to look over the Lusternik-Schnirelmann theory (L-S theory, for short) and the Topological Complexity (TC, for short) as a natural extension of the L-S theory. In particular, we focus on the impact of the ideas originated from E. Fadell and S. Husseini on both theories. More precisely, we see how their ideas on a category weight and a relative category drive the L-S theory and the TC.

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arXiv2021

Whitney Approximation for Smooth CW Complex

We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists a partition of unity subordinate to the open covering.

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arXiv2018

A short proof for tc(K) = 4

We show a method to determine topological complexity from the fibrewise view point, which provides an alternative proof for tc(K) = 4, where K denotes Klein bottle.

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arXiv2015

Mayer-Vietoris sequence for differentiable/diffeological spaces

The idea of a space with smooth structure is a generalization of an idea of a manifold. K. T. Chen introduced such a space as a differentiable space in his study of a loop space to employ the idea of iterated path integrals \cite{Chen:73,Chen:75,Chen:77,Chen:86}. Following the pattern established by Chen, J. M. Souriau \cite{Souriau:80} introduced his version of a space with smooth structure, which is called a diffeological space. These notions are strong enough to include all the topological spaces. However, if one tries to show de Rham theorem, he must encounter a difficulty to obtain a partition of unity and thus the Mayer-Vietoris exact sequence in general. In this paper, we introduce a new version of differential forms to obtain a partition of unity, the Mayer-Vietoris exact sequence and a version of de Rham theorem in general. In addition, if we restrict ourselves to consider only CW complexes, we obtain de Rham theorem for a genuine de Rham complex, and hence the genuine de Rham cohomology coincides with the ordinary cohomology for a CW complex.

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arXiv2013

On Lusternik-Schnirelmann category of SO(10)

Let $G$ be a compact connected Lie group and $p : E\to ΣA$ be a principal G-bundle with a characteristic map $α: A\to G$, where $A=ΣA_{0}$ for some $A_{0}$. Let $\{K_{i}{\to} F_{i-1}{\hookrightarrow} F_{i} \,|\, 1{\le} i {\le} n,\, F_{0}{=} \{\ast\} \; F_{1}{=} Σ{K_{1}} \; \text{and}\; F_{n}{\simeq} G \}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=Σ{K'_{1}} \subset F_{1}$ with $K'_{1} \subset K_{1}$ which satisfy $F_{i}F'_{1} \subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\operatorname{cat}(X) \le m{+}1$, if firstly the characteristic map $α$ is compressible into $F'_{1}$, secondly the Berstein-Hilton Hopf invariant $H_{1}(α)$ vanishes in $[A, ΩF'_1{\ast}ΩF'_1]$ and thirdly $K_{m}$ is a sphere. We apply this to the principal bundle $\mathrm{SO}(9)\hookrightarrow\mathrm{SO}(10)\to S^{9}$ to determine L-S category of $\mathrm{SO}(10)$.

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arXiv2012

Topological Complexity is a Fibrewise L-S Category

Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version $\TCM{B}$ of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities $\TC{B} = \catBb{\double{B}}+1$ and $\TCM{B} = \catBB{\double{B}}+1$, where $\double{B}=B{\times}B$ is a fibrewise pointed space over $B$ whose projection and section are given by $p_{\double{B}}=\proj_{2} : B{\times}B \to B$ the canonical projection to the second factor and $s_{\double{B}}=Δ_{B} : B \to B{\times}B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres. Recently, we found a problem with the proof of Theorem 1.13 which states that for a fibrewise well-pointed space $X$ over $B$, we have $\catBB{X}$ = \catBb{X}$ and that for a locally finite simplicial complex $B$, we have $\TC{B} = \TCM{B}$. While we still conjecture that Theorem 1.13 is true, this problem means that, at present, no proof is given to exist. Alternatively, we show the difference between two invariants $\catBb{X}$ and $\catBB{X}$ is at most 1 and the conjecture is true for some cases. We give further corrections mainly in the proof of Theorem 1.12.

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arXiv2012

Co-H-spaces and almost localization

Apart from simply-connected spaces, a non simply-connected co-H-space is a typical example of a space X with a co-action of $Bπ_1(X)$ along $r^X : X \rightarrow Bπ_{1}(X)$ the classifying map of the universal covering. If such a space X is actually a co-H-space, then the fibrewise p-localization of $r^X$ (or the `almost' p-localization of X) is a fibrewise co-H-space (or an `almost' co-H-space, resp.) for every prime p. In this paper, we show that the converse statement is true, i.e., for a non simply-connected space X with a co-action of $Bπ_1(X)$ along $r^X$, X is a co-H-space if, for every prime p, the almost p-localization of X is an almost co-H-space.

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arXiv2005

Lusternik-Schnirelmann category of Spin{9}

Let G be a compact connected Lie group and p : E \to Σ^2V a principal G-bundle with a characteristic map α: A=ΣV \to G. By combining cone decomposition arguments in Iwase-Mimura-Nishimoto [3,5] with computations of higher Hopf invariants introduced in Iwase [8], we generalize the result in Iwase-Mimura [12]: Let {F_{i}|0 \leq i \leq m} be a cone-decomposition of G with a canonical structure map σ_{i} of cat(F_{i}) \leq i for i \leq m. We have cat(E) \leq \Max(m+n,m+2) for n \geq 1, if αis compressible into F_{n} \subseteq F_{m} \simeq G and H^{σ_n}_n(α) = 0, under a suitable compatibility condition. On the other hand, calculations of Hamanaka-Kono [3] and Ishitoya-Kono-Toda [5] on spinor groups yields a lower estimate for the L-S category of spinor groups by means of a new computable invariant Mwgt(-;{mathbb{F}_2}) which is stronger than wgt(-;{\mathbb{F}_2}) introduced in Rudyak [16] and Strom [18]. As a result, we obtain cat(Spin(9)) = Mwgt(Spin(9);\mathbb{F}_2) = 8 > 6 = wgt(Spin(9);\mathbb{F}_2).

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arXiv2004

Lusternik-Schnirelmann categories of non-simply connected compact simple Lie groups

Let $F \hookrightarrow X \to B$ be a fibre bundle with structure group $G$, where $B$ is $(d{-}1)$-connected and of finite dimension, $d \geq 1$. We prove that the strong L-S category of $X$ is less than or equal to $m + \frac{\dim B}{d}$, if $F$ has a cone decomposition of length $m$ under a compatibility condition with the action of $G$ on $F$. This gives a consistent prospect to determine the L-S category of non-simply connected Lie groups. For example, we obtain $\cat{PU(n)} \leq 3(n{-}1)$ for all $n \geq 1$, which might be best possible, since we have $\cat{\mathrm{PU}(p^r)}=3(p^r{-}1)$ for any prime $p$ and $r \geq 1$. Similarly, we obtain the L-S category of $\mathrm{SO}(n)$ for $n \leq 9$ and $\mathrm{PO}(8)$. We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category.

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arXiv2004

Splitting off Rational Parts in Homotopy Types

It is known algebraically that any abelian group is a direct sum of a divisible group and a reduced group (See Theorem 21.3 of \cite{Fuchs:abelian-group}). In this paper, conditions to split off rational parts in homotopy types from a given space are studied in terms of a variant of Hurewicz map, say $\barρ : [S_{\Q}^{n},X] \to H_n(X;\Z)$ and generalized Gottlieb groups. This yields decomposition theorems on rational homotopy types of Hopf spaces, $T$-spaces and Gottlieb spaces, which has been known in various situations, especially for spaces with finiteness conditions.

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arXiv2004

Implications of the Ganea Condition

Suppose the spaces X and X cross A have the same Lusternik-Schnirelmann category: cat(X cross A)= cat(X). Then there is a strict inequality cat(X cross (A halfsmash B)) < cat (X) + cat(A halfsmash B) for every space B, provided the connectivity of A is large enough (depending only on X). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres.

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arXiv2002

A_{\infty}-method in Lusternik-Schnirelmann category

To clarify the method behind the paper "Ganea's conjecture on Lusternik-Schnirelman category" by the author, a generalisation of Berstein-Hilton Hopf invariants is defined as `higher Hopf invariants'. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea's conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of "Ganea's ..." for all the cases except the case when $p=2$. As an application, conditions in terms of homotopy invariants of the characteristic maps are given to determine the LS category of sphere-bundles-over-spheres. Consequently, a closed manifold $M$ is found not to satisfy Ganea's conjecture on LS category and another closed manifold $N$ is found to have the same LS category as its `punctured submanifold' $N-\{P\}$, $P \in N$. But all examples obtained here support the conjecture in "Ganea's ...".

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arXiv2002

Lusternik-Schnirelmann category of a sphere-bundle over a sphere

A criterion to determine the L-S category of a total space of a sphere-bundle over a sphere is given in terms of homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for such a total space $N$ to have the same L-S category as its `once punctured submanifold' $N\smallsetminus\{P\}$, $P \in N$. Also a necessary condition for such a total space $M$ to satisfy Ganea's conjecture is obtained.

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