• Honam Mathematical Journal
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• v.38 no.4
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• pp.819-831
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• 2016

The present paper investigates some properties of the digital homology in [1, 4, 5] and points out some unclearness of the definition of a digital homology and further, suggests a suitable form of a digital homology. Finally, we calculate a digital homology group and a relative digital homology group of some digital wedge sums. Finally, the paper corrects some errors in [6]. In the present paper all digital images (X, k) are assumed to be non-empty and k-connected.

• Honam Mathematical Journal
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• v.20 no.1
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• pp.121-133
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• 1998

We find some properties of the pro-torsion products. Under the suitable conditions, we also show that the map ${\bar{H}}_P({\chi};G){\rightarrow}{\bar{H}}_p^{s(r)}({\chi};G)$ is an isomorphism and the n-th homotopy group of X is isomorphic to the n-th ${\check{C}}ECH$ homology group.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.699-709
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• 2008

We study homology of the gauge group associated with the principal $G_2$ bundle over the four-sphere using the Eilenberg-Moore spectral sequence and the Serre spectral sequence with the aid of homology and cohomology operations.

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.175-183
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• 2010

The classical group completion theorem states that under a certain condition the homology of ${\Omega}BM$ is computed by inverting ${\pi}_0M$ in the homology of M. McDuff and Segal extended this theorem in terms of homology fibration. Recently, more general group completion theorem for simplicial spaces was developed. In this paper, we construct a symmetric monoidal 2-category ${\mathcal{A}}$. The 1-morphisms of ${\mathcal{A}}$ are generated by three atomic 2-dimensional CW-complexes and the set of 2-morphisms is given by the group of path components of the space of homotopy equivalences of 1-morphisms. The main part of the paper is to compute the homotopy type of the group completion of the classifying space of ${\mathcal{A}}$, which is shown to be homotopy equivalent to ${\mathbb{Z}}{\times}BAut^+_{\infty}$.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.215-222
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• 2015

We show that homology group on a contact CR-warped product submanifold in odd dimensional sphere is zero under certain conditions in terms of warping function and the dimension of the submanifold.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.399-408
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• 2006

The action of mapping class groups of a handlebody on the first homology group is investigated. A homological analogy of mapping class groups of a handlebody is defined and its system of generators is given.

• Journal of the Korean Mathematical Society
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• v.35 no.1
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• pp.149-164
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• 1998

We study the homology of the tripe loop space of the exceptional Lie group $F_4$ by exploiting the spectral sequences and the homology operators.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1497-1522
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• 2023

In this paper we compute the Bredon homology of wallpaper groups with respect to the family of finite groups and with coefficients in the complex representation ring. We provide explicit bases of the homology groups in terms of irreducible characters of the stabilizers.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.63-75
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• 2006

We study the homology of based gauge groups associated with the principal Spin(n) bundles over the four-sphere using the Eilenberg-Moore spectral sequence and the Serre spectral sequence with the Dyer-Lash of operations.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.563-571
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• 1995

The recent work of Hopkins, Kuhn and Ravenel [H-K-R] indicates the Morava K-theory, $K(n)^*(-)$, occupy an important and fundamental place in homology theory. In particular $K(n)^*(BG)$ for classifying spaces of finite groups are studied by many authors [H-K-R], [R], [T-Y 1, 2] and [Hu].