Norio Iwase

Ganea's Conjecture on Lusternik–Schnirelmann Category

A series of complexes Qp indexed by all primes p is constructed with catQp =2 and catQpS n =2f or eithern 2o rn =1a ndp = 2. This disproves Ganea’s conjecture on LS category, or Lusternik-Schnirelmann category.

A∞-method in Lusternik-Schnirelmann category

Abstract To clarify the method behind (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634), 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 Ganeas conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634) 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 Ganeas conjecture on LS category and another closed manifold N is found to have the same LS category as its ‘punctured submanifold’ N−{P}, P∈N . But all examples obtained here support the conjecture in (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634).

Topological complexity is a fibrewise L-S category

Abstract 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 TC M ( 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 ) = cat B ∗ ( d ( B ) ) + 1 and TC M ( B ) = cat B B ( d ( B ) ) + 1 , where d ( B ) = B × B is a fibrewise pointed space over B whose projection and section are given by p d ( B ) = pr 2 : B × B → B the canonical projection to the second factor and s d ( B ) = Δ B : B → B × B the diagonal. In addition, our method in studying fibrewise L–S category is able to treat a fibrewise space with singular fibres.

L-S Categories of Simply-connected Compact Simple Lie Groups of Low Rank

We determine the L-S category of Sp(3) by showing that the 5-fold reduced diagonal Δ5 is given by V 2using a Toda bracket and a generalised cohomology theory h* given by h* ( X, A ) = { X/A,S [0,2] }, where S[0, 2] is the 3-stage Postnikov piece of the sphere spectrum S. This method also yields a general result that cat (Sp(n)) > n + 2 for n > 3, which improves a result of Singhof [20].

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

Abstract We determine the Lusternik–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 Ganeas 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∈N . Also, necessary and sufficient conditions for a total space M to satisfy Ganeas conjecture are described.

Lusternik-schnirelmann category of spin(9)

First we give an upper bound of cat (E), the L-S category of a principal G-bundle E for a connected compact group G with a characteristic map a: ΣV → G. Assume that there is a cone-decomposition {F i | 0 1, if a is compressible into F n C F m ≃ G with trivial higher Hopf invariant H n (α). Second, we introduce a new computable lower bound, Mwgt(X; F 2 ) for cat(X). The two new estimates imply cat(Spin(9)) = Mwgt(Spin(9);F 2 ) = 8 > 6 = wgt(Spin(9);F 2 ), where (wgt-; R) is a category weight due to Rudyak and Strom.

A continuous localization and completion

The main goal of this paper is to construct a localization and completion of Bousfield-Kan type as a continuous functor for a virtually nilpotent CW-complex. Then the localization and completion of an An-space is given to be an An-homomorphism between An-spaces. For any general compact Lie group, this gives a continuous equivariant localization and completion for a virtually nilpotent G-CW-complex. More generally, we have a continuous localization with respect to a system of core rings for a virtually nilpotent DCW-complex for a polyhedral category D. The simplicial construction of Bousfield and Kan [2] automatically generalizes to the homology localization of a D-CW-complex [6] for a discrete category D, including the equivariant case for finite groups. But, this fails for general compact Lie groups. A. Elmendorf [8] constructs functorially an equivariant Eilenberg-Mac Lane space by using Mays method and the iteration of a bar construction. Using this, May et al. [16, 17] show the existence of an equivariant localization and completion using Theorems 3 and 4 of [8] for nilpotent G-CW-complexes, which is given with the Arithmetic Square Theorem for nilpotent G-spaces. On the other hand, T. Sumi [24] gives an equivariant localization of a 1-connected G-CW-complex with respect to a system of local rings by using the method of [20]. But, the functors are not continuous. In this paper, we construct a generalized Eilenberg-Mac Lane space R(X) by using the symmetric product [4]. Then by the methods of a triple (an algebra functor [1]) and a cosimplicial space, we construct a nilpotent tower, and a completion and a localization as continuous functors. Using this, we show the Arithmetic Square Theorem for a virtually nilpotent CW-complex (see DrorDwyer-Kan [5]). As the localization is continuous, the localization of an An-space (mapping) is an An-space (mapping) by the explicit definition of an An-space in Stasheff [22] and an An-mapping between An-spaces in [12, 13]. Received by the editors February 19, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P60; Secondary 55N91, 55P20, 55U40.

On the cellular decomposition and the Lusternik-Schnirelmann category of Spin(7)

Abstract 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).

Co-H-spaces and the Ganea conjecture

Abstract 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→Bπ 1 (X) , which is a space with a co-action of Bπ1(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.

Homology of the universal covering of a co-H-space

The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-IS-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 coH-spaces in a slightly general situation. As a corollary, we get the Ganea conjecture for spaces up to dimension 3.