Norio Iwase | Last modified date:2024.03.22 |
Professor /
Division of Algebra and Geometry /
Faculty of Mathematics
Papers
1. | Norio Iwase, Smooth A∞-form on a diffeological loop space, Contemporary Mathematics, https://doi.org/10.1090/conm/794/15926, 794, 223-238, 2024.01, [URL], To construct an A∞-form for a loop space in the category of dif- feological spaces, we have two minor problems. Firstly, the concatenation of paths in the category of diffeological spaces needs a small technical trick, which apparently restricts the number of iterations of concatenations. Secondly, we do not know a 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∞-space. In appendix, we show an alternative solution by modifying the concatenation to be stable without assuming reflexivity on spaces nor stability on paths.. |
2. | Norio Iwase, Lusternik–Schnirelmann theory to topological complexity from A∞-view point, Topological Methods in Nonlinear Analysis, 10.12775/TMNA.2022.060, 61, 1, 217-238, 2023.03, 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.. |
3. | Norio IWASE, Yuya Miyata, Topological complexity of $S^3/Q_8$ as fibrewise L-S category, Topological Methods in Nonlinear Analysis, DOI:10.12775/TMNA.2022.068, 62, 1, 239-265, 2023.01, 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 $\proj_{1} \colon X \times X \to X$ with the diagonal map $\Delta \colon X \to X \times X$ 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.. |
4. | Norio Iwase, Whitney Approximation for Smooth CW Complex, Kyushu J. Math., 10.2206/kyushujm.76.177, 76, 1163, 197-200, 2022.02, We show a Whitney approximation theorem for a continuous map from a manifold to a smooth CW complex, which 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. In addition, we observe that there are enough many smooth functions on a smooth CW complex.. |
5. | Upper bound for monoidal topological complexity. |
6. | Norio IWASE, Mitsunobu Tsutaya, Upper Bound for Monoidal Topological Complexity, Kyushu Journal of Mathematics, doi:10.2206/kyushujm.74.197, 74, 1, 197-200, 2020.02, We show that tc^M(M) ≤ 2cat(M) for a finite simplicial complex M. For example, we have tc^M(S^n ∨ S^m) = 2 for any positive integers n and m.. |
7. | Norio Iwase, MIchihiro Sakai, Mitsunobu Tsutaya, A short proof for $\tc{K}=4$, Topology and its Applications, 10.1016/j.topol.2019.06.014, 264, 1, 167-174, 2019.12. |
8. | Norio Iwase, Nobuyuki Izumida, Mayer-Vietoris sequence for differentiable/diffeological spaces, Trends in Mathematics, 10.1007/978-981-13-5742-8_8, 123-151, Algebraic Topology and Related Topics (Mohali, 2017), 123--151, {\em Trends in Mathematics}, Springer Singapore, 2019., 2019.03, 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.. |
9. | Iwase, Norio; Kikuchi, Kai; Miyauchi, Toshiyuki, On Lusternik-Schnirelmann category of SO(10), Fundamenta Mathematicae, DOI: 10.4064/fm678-11-2015, 234, 3, 201-227, 2016.04, Let G be a compact connected Lie group and p: E → ΣA be a principal G-bundle with a characteristic map α: A → G, where A = ΣA₀ for some A₀. Let ${K_{i} → F_{i-1} ↪ F_{i} | 1 ≤ i ≤ m}$ with F₀ = {∗}, F₁ = ΣK₁ and Fₘ ≃ G be a cone-decomposition of G of length m and F'₁ = ΣK'₁ ⊂ F₁ with K'₁ ⊂ K₁ which satisfy $F_{i}F'₁ ⊂ F_{i+1}$ up to homotopy for all i. Then cat(E) ≤ m + 1, under suitable conditions, which is used to determine cat(SO(10)). A similar result was obtained by Kono and the first author (2007) to determine cat(Spin(9)), but that result could not yield cat(E) ≤ m + 1.. |
10. | Norio Iwase, C. R. Costoya, Co-H-Spaces and almost localization, Proceedings of the Edinburgh Mathematical Society, DOI: http://dx.doi.org/10.1017/S0013091514000200, 58, 02, 323-332, 2015.06, Apart from simply connected spaces, a non-simply connected co-H-space is a typical example of a space X with a coaction of B pi(1)( X) along r(X) : X -> B pi(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, respectively) 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 coaction of B pi(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.. |
11. | N. Iwase, M. Mimura, N. Oda and Y. S. Yoon, The Milnor--Stasheff Filtration on Spaces and Generalized Cyclic Maps, Canadian Mathematical Bulletin, doi:10.4153/CMB-2011-130-8, 55, 3, 523-536, 2012.07. |
12. | Norio Iwase, MIchihiro Sakai, Erratum to ``Topological complexity is a fibrewise L-S category'', Topology and its Applications, 159, 10-11, 2810-2813, 2012.05. |
13. | Norio Iwase, MIchihiro Sakai, Topological complexity is a fibrewise L-Scategory, Topology and its Applications, 10.1016/j.topol.2009.04.056, http://dx.doi.org/10.1016/j.topol.2009.04.056, 2010.05. |
14. | Norio Iwase, Categorical length, relative L-Scategory and higher Hopf invariants, Algebraic Topology: Old and New (Bedlewo, 2007), Banach Center Publ., 85(2009), 205–224, 2009.06. |
15. | Norio Iwase, Michihiro Sakai, Functors on the category of quasi-fibrations, Topology and its Applications, 155 (2008), 1403--1409, 2008.06. |
16. | Norio Iwase, Akira Kono, Lusternik-Schnirelmann category of Spin(9), Transactions of the American Mathematical Society, 359 (2007), 1517–1526, 2007.06. |
17. | Norio Iwase, Toshiyuki Miyauchi, Lusternik-Schnirelmann category of stunted quasi-projective spaces, Journal of Mathematics of Kyoto University, 47 (2007), 321–326, 2007.06. |
18. | Norio Iwase, Mamoru Mimura, Tetsu Nishimoto, Lusternik-Schnirelmann category of non-simply connected compact simple Lie groups, Topology and its Applications, 10.1016/j.topol.2004.11.006, 150, 1-3, 111-123, 2005.06. |
19. | Norio Iwase, Nobuyuki Oda, Splitting off rational parts in homotopy types, Topology and its Applications, 10.1016/j.topol.2005.01.027, 153, 1, 133-140, 153 (2005), no. 1, 133--140, 2005.06. |
20. | Norio Iwase, Donald Stanley, Jeffrey Strom, Implications of the Ganea condition, Algebraic and Geometric Topology, 4 (2004), 829–839, 2004.06, Suppose the spaces $X$ and $X \times A$ have the same Lusternik-Schnirelmann category: $\cat(X \times A) = cat(X)$. Then there is a strict inequality $\cat(X \times (A \rtimes B)) |
21. | Norio Iwase, Mamoru Mimura, L-S categories of simply-connected compact simple Lie groups of low rank, Progress Math., 215, 199-212, 215, 199--212, 2004.01. |
22. | Norio Iwase, Lusternik-Schnirelmann category of a sphere-bundle over a sphere, Topology, 10.1016/S0040-9383(02)00026-5, 42, 3, 701-713, 42, 701--713, 2003.01, We determine the Lustemik-Schnirelmann (L-S) category of a total space of a sphere-bundle over a sphere in terms of primary 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 a total space N to have the same L-S category as its 'once punctured submanifold' N{P}, P is an element of N. Also, necessary and sufficient conditions for a total space M to satisfy Ganea's conjecture are described. . |
23. | Norio Iwase, Mamoru Mimura, Tetsu Nishimoto, On the cellular decomposition and the Lusternik-Schnirelmann category of Spin(7), Topology and its application, 10.1016/S0166-8641(03)00038-5, 133, 1, 1-14, 133, 1--14, 2003.01, We give a cellular decomposition of the compact connected Lie group Spin(7). We also determine the L-S categories of Spin(7) and Spin(8).. |
24. | Norio Iwase, A_infinity-method in Lusternik-Schnirelmann category, Topology, 10.1016/S0040-9383(00)00045-8, 41, 4, 695-723, 41, 695--723, 2002.01, Berstein-Hilton Hopf invariants are generalised to detect the higher homotopy associativity of a Hopf space as `higher Hopf invariants', which are studied as obstructions for normalised Lusternik-Schnirelmann category, LS category for short. Under a condition among dimension and LS category, the criterion for Ganea's conjecture on LS category is obtained using the stabilised higher Hopf invariants and the conjecture in "Ganea's conjecture on Lusternik-Schnirelmann category" is verified, which yields the main result in it except the case when p=2. As an application, conditions in terms of homotopy invariants of the attaching maps are given to determine LS category of sphere-bundles-over-spheres: A closed manifold is found to have the same LS category as its punctured submanifold and another closed manifold is found not to satisfy Ganea's conjecture on LS category.. |
25. | John Hubbuck, Norio Iwase, A p-complete version of the Ganea conjecture on co-H-spaces, Contemp. Math., 316, 127--133, 2002.01, A finite connected CW complex which is a co-H-space is shown to have the homotopy type of a wedge of a bunch of circles and a simply-connected finite complex after almost $p$-completion at a prime $p$.. |
26. | Norio Iwase, Co-H-spaces and the Ganea conjecture, Topology, 10.1016/S0040-9383(99)00052-X, 40, 2, 223-234, 40, 223--234, 2001.01, The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-H-space, a space with co-H-structure. Many efforts are devoted to show the Ganea conjecture under additional assumptions on the given co-H-structure. In this paper, we construct a series of co-H-spaces, each of which cannot be split into a one-point-sum of a simply connected space and a bunch of circles, disproving the Ganea conjecture: a non-simply connected co-H-space X is, up to homotopy, the total space of a fibrewise-simply connected pointed fibrewise co-Hopf fibrant j:X --> B pi (1)(X), which is a space with a co-action of B pi (1)(X) along j. We construct its homology decomposition, which yields a simple construction of its fibrewise localisation. Using the fact that 9^2=9 and (-8)^2=-8 mod 24 together with 9 + (-8)=1, we obtain the result.. |
27. | Hans Baues, Norio Iwase, Square rings associated to elements in homotopy groups of spheres, Contemp. Math., 274, 57--78, 2001.01. |
28. | Norio Iwase, Shiroshi Saito, Toshio Sumi, Homology of the universal covering of a co-H-space, Trans. Amer. Math. Soc., 10.1090/S0002-9947-99-02238-2, 351, 12, 4837-4846, 351, 4837-4846, 1999.01, The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-H-space, a space with co-H-structure. Many efforts are devoted to show the Ganea conjecture under additional assumptions on the given co-H-structure. In this paper, we show a homological property of co-H-spaces in a slightly general situation. As a corollary, we get the Ganea conjecture for spaces up to dimension 3.. |
29. | Norio Iwase, Akira Kono, Mamoru Mumura, Adjoint action of a finite loop space II, Roy. Soc. Edinburgh, 129, 773-785, 129, 773-785, 1999.01. |
30. | Norio Iwase, Ganea's conjecture on Lusternik-Schnirelmann category, Bull. London Math. Soc., 10.1112/S0024609398004548, 30, 623-634, 30, 623-634, 1998.01, The problem 2 posed by Tudor Ganea is known as the Ganea conjecture on Lusternik-Schnirelmann category, or the "Ganea Conjecture". Many efforts are devoted to show the Ganea conjecture under additional assumptions on a space. In this paper, we construct a series of spaces indexed by primes, which disproves the "Ganea conjecture". The method behind the result given in this paper is given in a separate paper published in Topology in 2002.. |
31. | Norio Iwase, Adjoint action of a finite loop space, Proc. Amer. Math. Soc., 10.1090/S0002-9939-97-03924-5, 125, 9, 2753-2757, 125, 2753-2757, 1997.01. |
32. | Norio Iwase, Certain missing terms in an unstable Adams spectral sequence, Mem. Fac. Sci. Kyu. U. Math., 41, 97--113, 1987.01. |
33. | Norio Iwase, H-spaces with generating subspaces, Proc. Roy. Soc. Edinburgh, 111, 199-211, 111A, 199--211, 1989.01. |
34. | Norio Iwase, Homotopy associativity of sphere extensions, Proc. Edinburgh Math. Soc., 32, 459--472, 1989.01. |
35. | Norio Iwase, Mamoru Mimura, Higher homotopy associativity, Proceeding of Arcata conference, Lec. Not. Math., 1370, 193-220, 1370, 193--220, 1989.01. |
36. | Norio Iwase, A continuous localization and completion, Trans. Amer. Math. Soc., 320, 77--90, 1990.01. |
37. | Norio Iwase, Kenichi Maruyama, Shichiro Oka, A note on E (HP^n) for n = 4, Math. J. Okayama U. 33, 163-175, 33, 163--175, 1991.01. |
38. | Norio Iwase, Akira Kono, Mamoru Mumura, Generalized Whitehead spaces with few cells, Publ. Res. Inst. Math. Sci. Kyo. U., 28, 615-652, 1992.01. |
39. | Norio Iwase, On the splitting of mapping spaces between classifying spaces I, Publ. Res. Inst. Math. Sci. Kyoto U., 10.2977/prims/1195176439, 23, 3, 445-453, 23, 445--453, 1987.01. |
40. | Norio Iwase, On the $K$-ring structure of $X$-projective $n$-space, Mem. Fac. Sci. Kyu. U. Math., 10.2206/kyushumfs.38.285, 38, 2, 285-297, 1984.01. |
Unauthorized reprint of the contents of this database is prohibited.