Kyushu University Academic Staff Educational and Research Activities Database
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Norio Iwase Last modified date:2024.03.02



Graduate School
Undergraduate School


Homepage
https://kyushu-u.elsevierpure.com/en/persons/norio-iwase
 Reseacher Profiling Tool Kyushu University Pure
http://www2.math.kyushu-u.ac.jp/~iwase/index-e.html
Academic Degree
DSc at Kyushu University
Country of degree conferring institution (Overseas)
No
Field of Specialization
Topology
Total Priod of education and research career in the foreign country
02years03months
Outline Activities
Research Activity:
The number of critical points of a smooth function from a manifold M is bounded by a homotopy-theoretical invariant cat(M) the Lusternik-Schnirelmann category. Similarly to it, M. Farber introduced an invariant called Topological Complexity measuring how a space is complex, which opened the door to apply algebraic topology to the world outside mathematics. Quite recently, I am interested in Chen-Souriau differentiable spaces which enables us to introduce differentiable structures in every topological spaces. Currently, I am investigating homotopy-theoretical properties of L-S category and Topological Complexity by calculating L-S categories for Lie groups and total spaces of fibre bundles.
Education Activity:
Lectures on linear algebra and/or Analysis for first and/or second year classes, and lectures on Topology for graduate and/or undergraduate classes. Sometimes are given. Seminar courses are sometimes given for graduate and/or undergraduate students.
Research
Research Interests
  • de Rham theory on diffeological spaces
    keyword : Diffeological space
    2013.10.
  • Topological Complexity of Configuration Spaces of Robot Arms
    keyword : Topological Complexity
    2008.02.
  • Homotopy Theory from the categorical point of view
    keyword : square ring, 2-category
    2002.04Homotopy theory from the category-theoretical view point.
  • L-S category of a manifold
    keyword : Lusternik-Schnirelmann category
    2000.04L-S category of manifolds.
  • Ganea conjecture for a co-Hopf space
    keyword : Ganea conjecture, co-Hopf space
    1990.04The Ganea conjecture on co-Hopf spaces.
Academic Activities
Reports
1. The Ganea conjecture and recent progress on Lusternik-Schnirelmann category
[translation of Sūgaku 56 (2004), 281–296], Sugaku Expositions, 20(2007)..
Papers
1. Norio Iwase, Akira Kono, Lusternik-Schnirelmann category of Spin(9), Transactions of the American Mathematical Society, 359 (2007), 1517–1526, 2007.06.
2. 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.
3. 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.
4. 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. .
5. 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..
6. 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..
7. 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..
8. Norio Iwase, H-spaces with generating subspaces, Proc. Roy. Soc. Edinburgh, 111, 199-211, 111A, 199--211, 1989.01.
9. Norio Iwase, Mamoru Mimura, Higher homotopy associativity, Proceeding of Arcata conference, Lec. Not. Math., 1370, 193-220, 1370, 193--220, 1989.01.
10. 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.
Works, Software and Database
1. .
Presentations
1. Lusternik-Schnirelmann category.
2. Lusternik-Schnirelmann category and an A_infinity structure.
3. Norio Iwase, Recent progress on Lusternik-Schnirelmann category, International Conference on Topology and its Applications Joined with 2nd Japan-Mexico Topology Symposium "Topology in Matsue", 2002.06.
4. The Ganea conjecture and its counter examples.
5. Norio Iwase, Some problems by Ganea and their p-localised versions, JAMI (The Japan-U.S. Mathematics Institute) 2000 Program "Recent Progress in Homotopy Theory", 2000.03.
6. A_infinity-method in L-S category.
Educational
Educational Activities
My Education activities often (not always) includes the following four.
1. Supervising graduate students in master and/or doctor courses.
2. Supervising 3rd-year and/or 4th-year undergraduate students.
3. Basic and/or advanced classes on Topology.
4. Lectures and/or tutorial classes on first and/or second year undergraduate students.
Social
Professional and Outreach Activities
Extension Lecturer at Faculty of Mathematics, Kyushu University (July, 2006)
Member of the Committee of Extension Course of Faculty of Mathematics, Kyushu University (Apr. 2008 - Mar. 2018).