Nobuyuki Oda

SOME APPLICATIONS OF MINIMAL OPEN SETS

We characterize minimal open sets in topological spaces. We show that any nonempty subset of a minimal open set is pre-open. As an application of a theory of minimal open sets, we obtain a sufficient condition for a locally finite space to be a pre- Hausdorff space. 2000 Mathematics Subject Classification. 54A05, 54D99. 1. Introduction. Let X be a topological space. We call a nonempty open set U of X a minimal open set when the only open subsets of U are U and ∅. In this paper, we study fundamental properties of minimal open sets and apply them to obtain some results on pre-open sets (cf. (2)) and pre-Hausdorff spaces. In Section 2, we characterize minimal open sets, that is, we show that a nonempty open set U is a minimal open set if and only if Cl(U) = Cl(S) for any nonempty subset S of U. This result implies that any nonempty subset S of a minimal open set U is a pre-open set. In Section 3, we study minimal open sets in locally finite spaces. The results of this section are closely related to the work of James (1), and these results will be used in the next scetion. In Section 4, we apply the theory of minimal open sets to study pre-open sets. Our first main result of this section is a property of the set of all minimal open sets in any nonempty finite open set which is not a minimal open set. This result enables us to prove a generalization of Theorem 2.5, when U is a nonempty finite open set, in Theorem 4.4. Theorem 4.5 shows that our theory of minimal open set is useful to study pre-open sets. Finally, we show that some conditions on minimal open sets implies pre-Hausdorff- ness of a space, that is, if any minimal open set of a locally finite space X has two elements at least, then X is a pre-Hausdorff space.

Some properties of maximal open sets

Some fundamental properties of maximal open sets are obtained, such as decomposition theorem for a maximal open set. Basic properties of intersections of maximal open sets are established, such as the law of radical closure.

Localization, completion and detecting equivariant maps on skeletons

In the category of equivariant spaces with base point, we prove the injectivity of the induced map between homotopy sets under some conditions. We study some relations between the localization and the completion. By using these results, we characterize continuous maps which are homotopic on skeletons, and obtain a generalization of the theory of phantom maps.

Minimal closed sets and maximal closed sets

Some properties of minimal closed sets and maximal closed sets are obtained, which are dual concepts of maximal open sets and minimal open sets, respectively. Common properties of minimal closed sets and minimal open sets are clarified; similarly, common properties of maximal closed sets and maximal open sets are obtained. Moreover, interrelations of these four concepts are studied.

Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.

A Γ-Whitehead product for track groups and its dual

In this paper, we generalize the product defined by Hardie and Jansen in [3]. The generalized product gives a pairing [·, ·] W Γ : [ΓX ∧ W, Z] × [ΓY ∧ W, Z] → [Γ(X ∧ Y) ∧ W, Z]. When Γ = S 1, X = S m−1 and Y = S n−1, it is the product of Hardie and Jansen. We study properties of it such as bilinearity, Jacobi identity and some formulas in [4]. Furthermore, we consider the dual product and investigate various properties corresponding under the principle of Eckmann-Hilton duality.

On the unstable homotopy spectral sequences

We shall study the relation between the unstable homotopy spectral sequence associated with the cell decomposition of the source space and the one associated with the Postnikov decomposition of the target space. It is proved that the Maunders result holds for these two types of spectral sequences. This result will be exploited to study the differentials of these spectral sequences and we obtain a generalization of a theorem due to Atiyah-Hirzebruch and Dold. Making use of these spectral sequences with the convergence lemma, some results on the phantom maps and the homotopy groups of map*(Y, X) will be proved.

The Milnor-stasheff filtration on spaces and generalized cyclic maps

The concept ofCk-spaces is introduced, situated at an intermediate stage between H-spaces and T-spaces. The Ck-space corresponds to the k-th Milnor–Stasheff filtration on spaces. It is proved that a space X is a Ck-space if and only if the Gottlieb set G(Z,X) = [Z,X] for any space Z with cat Z ≤ k, which generalizes the fact that X is a T-space if and only if G(ΣB,X) = [ΣB,X] for any space B. Some results on theCk-space are generalized to theC f k -space for a map f : A → X. Projective spaces, lens spaces and spaces with a few cells are studied as examples ofCk-spaces, and non-Ck-spaces.

GOTTLIEB GROUPS AND SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES

Let E#(X) be the subgroups of E(X) consisting of ho- motopy classes of self-homotopy equivalences that flx homotopy groups through the dimension of X and E⁄(X) be the subgroup of E(X) that flx homology groups for all dimension. In this pa- per, we establish some connections between the homotopy group of X and the subgroup E#(X) \ E⁄(X) of E(X). We also give some relations between …n(W), as well as a generalized Gottlieb group G f(W;X), and a subset M fN (X;W) of (X;W). Finally we es- tablish a connection between the coGottlieb group of X and the subgroup of E(X) consisting of homotopy classes of self-homotopy equivalences that flx cohomology groups.

Long Box Bracket Operations in Homotopy Theory

Secondary homotopy operations called box bracket operations were defined in the homotopy theory of an arbitrary 2-category with zeros by Hardie, Marcum and Oda (Rend Ist Mat Univ Trieste, 33:19–70 2001). For the topological 2-category of based spaces, based maps and based track classes of based homotopies, the classical Toda bracket is a particular example of a box bracket operation and subsequent development of the theory has refined, clarified and placed in this more general context many of the properties of classical Toda brackets. In this paper, and for the topological case only, we use an inductive definition to extend the theory to long box brackets. As is well-known, the necessity to manage higher homotopy coherence is a complicating factor in the consideration of such higher order operations. The key to our construction is the definition of an appropriate triple box bracket operation and consequently we focus primarily on the properties of the triple box bracket. We exhibit and exploit the relationship of the classical quaternary Toda bracket to the triple box bracket. As our main results we establish some computational techniques for triple box brackets that are based on composition methods. Some specific computations from the homotopy groups of spheres are included.