arXiv subjects

Dae-Woong Lee

Publications and source records attributed to Dae-Woong Lee.

7 recordsLinked to original sources

arXiv2025

Homotopy equivalence of digital pictures in $\mathbb{Z}^2$

We investigate the properties of digital homotopy in the context of digital pictures $(X,κ,\bar κ)$, where $X\subsetneq \Z^n$ is a finite set, $κ$ is an adjacency relation on $X$, and $\bar κ$ is an adjacency relation on the complement of $X$. In particular we focus on homotopy equivalence between digital pictures in $\Z^2$. We define a numerical homotopy-type invariant for digital pictures in $\Z^2$ called the outer perimeter, which is a basic tool for distinguishing homotopy types of digital pictures. When a digital picture has no holes, we show that it is homotopy equivalent to its rc-convex hull, obtained by ``filling in the gaps'' of any row or column. We show that a digital picture $(X,c_i,c_j)$ is homotopy equivalent to only finitely many other digital pictures $(Y,c_i,c_j)$. At the end of the paper, we raise a conjecture on the largest digital picture of the same homotopy-type of a given digital picture.

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arXiv2024

On digital H-spaces

In this article, we investigate properties of digital H-spaces in the graph theoretic model of digital topology. As in prior work, the results obtained often depend fundamentally on the choice between NP$_1$ and NP$_2$ product adjacencies. We explore algebraic properties of digital H-spaces preserved under digital homotopy equivalence, and we give a general construction that produces examples of digital H-spaces which are not homotopy-equivalent to digital topological groups in both categories. Further, we show that this construction essentially classifies all NP$_2$-digital H-spaces. In a short appendix, we resolve a question that was left unresolved in [17], and complete the full classification of digital topological groups.

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arXiv2022

Digital topological groups

In this article, we develop the basic theory of digital topological groups. The basic definitions directly lead to two separate categories, based on the details of the continuity required of the group multiplication. We define $\NP_1$- and $\NP_2$-digital topological groups, and investigate their properties and algebraic structure. The $\NP_2$ category is very restrictive, and we give a complete classification of $\NP_2$-digital topological groups. We also give many examples of $\NP_1$-digital topological groups. We define digital topological group homomorphisms, and describe the digital counterpart of the first isomorphism theorem.

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arXiv2020

Digital (co)homology modules and digital Pontryagin algebras

In the current study, we explore digital homology and cohomology modules, and investigate their fundamental properties on pointed digital images. We also examine pointed digital Hopf spaces and base point preserving digital Hopf functions between the pointed digital Hopf spaces with suitable digital multiplications, and explore the digital primitive homology and cohomology classes, the digital Pontryagin algebras and coalgebras on the digital Hopf spaces as digital images.

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arXiv2018

The same $n$-type structure of the suspension of the wedge products of the Eilenberg-MacLane spaces

For a connected CW-complex, we let $SNT(X)$ be the set of all homotopy types $[Y]$ such that the Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ of $X$ and $Y$, respectively, are homotopy equivalent for all positive integers $n$. In 1992, McGibbon and Møller (\cite[page 287]{MM}) raised the following question: Is $SNT(Σ\mathbb C P^\infty) = *$ or not? In this article, we give an answer to the more generalized version of this query: The set of all the same $n$-types of the suspended wedge sum of the Eilenberg-MacLane spaces of various types of both even and odd integers is the set which consists of only one element as a single homotopy type of itself.

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arXiv2017

The representation-ring-graded local cohomology spectral sequence for BPR<3>

The purpose of this paper is to examine the calculational consequences of the duality for BPR proved by the first author and Meier in 1607.02332, making them explicit in the special case n=3. We give an explicit description of the behaviour of the spectral sequence, with pictures.

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arXiv2010

On the same $N$-type of the suspension of the infinite quaternionic projective space

Let $[ρ_{i_k},[ρ_{i_{k-1}},...,[ρ_{i_{1}}, ρ_{i_2}] ...]]$ be an iterated commutator of self-maps $ρ_{i_j} : Σ{\Bbb H}P^\infty \to Σ{\Bbb H}P^\infty, j = 1,2, ..., k$ on the suspension of the infinite quaternionic projective space. In this paper, it is shown that the image of the homomorphism induced by the adjoint of this commutator is both primitive and decomposable. The main result in this paper asserts that the set of all homotopy types of spaces having the same $n$-type as the suspension of the infinite quaternionic projective space is the one element set consisting of a single homotopy type. Moreover, it is also shown that the group $\text{Aut}(π_{\leq n} (Σ{\Bbb H}P^\infty)/\text{torsion})$ of automorphisms is finite for $n \leq 9$, and infinite for $n \geq 13$, and that $\text{Aut}(π_{*} (Σ{\Bbb H}P^\infty)/\text{torsion})$ becomes non-abelian.

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