Kyushu University [Norio Iwase (Professor) Faculty of Mathematics, Department of Mathematics]

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

Books

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Reports

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1.	The Ganea conjecture and recent progress on Lusternik-Schnirelmann category [translation of Sūgaku 56 (2004), 281–296], Sugaku Expositions, 20(2007)..

Papers

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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.
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, 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. Our main result is the construction of 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, thus disproving the Ganea conjecture. .
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 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..
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

Presentations

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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 (April, 2008 - ).

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