Kyushu University Academic Staff Educational and Research Activities Database
List of Papers
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.