Katsuya Eda

The fundamental groups of one-dimensional spaces

Abstract Let X be a space of dimension at most 1. Then, the fundamental group is isomorphic to a subgroup of the first Cech homotopy group based on finite open covers. Consequently, for a one-dimensional continuum X , the fundamental group is isomorphic to a subgroup of the first Cech homotopy group.

A characterization of ℵ1-free abelian groups and its application to the chase radical

A groupA is an ℵ1-free abelian group iffA is a subgroup of the Boolean power Z(B) for some complete Boolean algebraB. The Chase radicalvA=Σ{C≦A: Hom(C, Z)=0 &C is countable). The torsion class {A:vA=A} is not closed under uncountable direct products.

Free σ-products and fundamental groups of subspaces of the plane

Let H be the so-called Hawaiian earring, i.e., H = {(x,y): (x − 1n)2 + y2 = 1n2, 1 ⩽ n < ω} and o = (0,0). We prove: 1. (1) If Y is a subspace of a line in the Euclidean plane R2 and X its complement R2β Y with x ϵ X, then the fundamental group π1(X, x) is isomorphic to a subgroup of π1(H, o). 2. (2) Let Y be a subspace of a line in the Euclidean plane R2. Then, π1(R2 β Y, x) for x ϵ R2 β Y is isomorphic to π1(H, o), if and only if there exists infinitely many connected components of Y which converge to a point outside of Y. 3. (3) Every homomorphism from π1(H, o) to itself is conjugate to a homomorphism induced from a continuous map.

Boolean powers of abelian groups

where Z is the group of the integers. This kind of group was first (1962) studied by Balcerzyk [l]. However, it seems that not much attention was paid to such groups for a rather long period. Under the point of view in [l, Theorem 51 and [13, Proposition 11, it can be said that there has been studies about it after around 1980 [5-18,24,29,30]. In the present paper we investigate cotorsion-freeness and algebraical compact- ness of Boolean powers. Undefined notions about abelian groups and Boolean algebras are the usual ones and can be found in [20] and [23] respectively. All groups in this paper are abelian groups.

Chapter 12 N-Compactness and Its Applications

Publisher Summary This chapter discusses N -compactness and its application and N -compact spaces. A real-compact space is a topological space that can be embedded in the product of copies of the real-line ℝ as a closed subspace. A topological space is called “ N -compact,” if it is homeomorphic to a closed subspace of the product of copies of the countable discrete space N . The chapter also introduces the notion of E -compact spaces, rings and lattices of continuous functions, and applications to Abelian groups and non-Archimedean Banach spaces. N -compact spaces can be regarded as a 0-dimensional analog of real-compact spaces. The chapter considers the relationship between real-compact spaces and N -compact spaces and gives technical results and examples. The chapter explains that the ring structure or the lattice structure of C ( X , ℤ) determines the topology of an N -compact space X , where C ( X , ℤ) is the ring or the lattice of integer-valued, continuous functions on X . Applications to Abelian groups, where N -compactness will play an important role to reduce the reflexivity of Abelian groups. Banach spaces over certain non-Archimedean-valued fields have many features on the reflexivity that are similar to those of Abelian groups. Applications to non-Archimedean Banach spaces apply N -compact spaces to such Banach spaces, with particular attention paid to their similarity.

Sequential fans in topology

Abstract For an index set I , let S ( I ) be the sequential fan with I spines, i.e., the topological sum of I copies of the convergent sequence with all nonisolated points identified. The simplicity and the combinatorial nature of this space is what lies behind its occurrences in many seemingly unrelated topological problems. For example, consider the problem which ask us to compute the tightness of the square of S ( I ). We shall show that this is in fact equivalent to the well-known and more crucial topological question of W. Fleissner which asks whether, in the class of first countable spaces, the property of being collectionwise Hausdorff at certain levels implies the same property at higher levels. Next, we consider Kodamas question whether or not every Σ-product of Lasnev spaces is normal. The sequential fan again enters the scene as we show S ( ω 2 ) × S ( ω 2 ) × ω 1 , which can be embedded in a Σ-product of Lasnev spaces as a closed set, can be nonnormal in some model of set theory. On the other hand, we show that the Σ-product of arbitrarily many copies of the slightly smaller fan S ( ω 1 ) is normal.

Prime subspaces in free topological groups

Abstract Let F(X) (A(X)) be the free (Abelian) topological group over X. We prove: If P is one of the spaces R , Q , R , Q , βω, βω ω and 2/gk for an infinite κ and if F(X) or A(X) contains a copy of P, then X contains a copy of P. If P is the one-point compactification of an infinite discrete space or ω1 + 1, this is not true. If P = ω1, this holds for F(X) but is independent of ZFCfor A(X).

On (co)homology locally connected spaces

We prove that there exists a cohomology locally connected compact metrizable space which is not homology locally connected. In the category of compact Hausdorff spaces a similar result was proved earlier by G.E. Bredon.  2002 Elsevier Science B.V. All rights reserved. AMS classification:Primary 55N05; 55N10, Secondary 20F12; 55N40; 55N99

The fundamental groups of one-dimensional wild spaces and the Hawaiian earring

Let X be a one-dimensional space which contains a copy C of a circle and let it not be semi-locally simply connected at any point on C. Then the fundamental group of X cannot be embeddable into a free σ-product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.

The effect of curvature on the instability of a solid/liquid interface

An essential factor causing the instability (stability) of a solid/liquid interface during solidification is explored. By examining the qualitative properties of the classical solution of a nonlinear evolution equation, the bifurcation, the coalescence and the growth rate of the interface are discussed. These discussions lead to the relation between the instability (stability) and the curvature of the interface.