• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.111-137
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• 2021

Higson proved that every homotopy invariant, stable and split exact functor from the category of C⁎-algebras to an additive category factors through Kasparov's KK-theory. By adapting a group equivariant generalization of this result by Thomsen, we generalize Higson's result to the inverse semigroup and locally compact, not necessarily Hausdorff groupoid equivariant setting.

• Bulletin of the Korean Mathematical Society
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• v.40 no.1
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• pp.17-28
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• 2003

Let X be a closed, oriented, Riemannian 4-manifold with ${{b_2}^+}(x)\;>\;1$ and of simple type. Suppose that ${\sigma}\;:\;X\;{\rightarrow}\;X$ is an involution preserving orientation with an oriented, connected, compact 2-dimensional submanifold $\Sigma$ as a fixed point set with ${\Sigma\cdot\Sigma}\;{\geq}\;0\;and\;[\Sigma]\;{\neq}\;0\;{\in}\;H_2(X;\mathbb{Z})$. We show that if _X(\Sigma)\;+\;{\Sigma\cdots\Sigma}\;{\neq}\;0$ then the $Spin^{C}$ bundle $\={P}$ is not $\mathbb{Z}_2-equivariant$, where det $\={P}\;=\;L$ is a basic class with $c_1(L)[\Sigma]\;=\;0$.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.219-224
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• 2015

In this paper it is investigated as to when a nonempty invariant closed subset A of a $S^1$-space X containing the set of stationary points (S) can be the fixed point set of an equivariant continuous selfmap on X and such space X is said to possess the S-equivariant complete invariance property (S-ECIP). It is also shown that if X is a metric space and $S^1$ acts on $X{\times}S^1$ by the action $(x,p){\cdot}q=(x,p{\cdot}q)$, where p, $q{\in}S^1$ and $x{\in}X$, then the hyperspace $2^{X{\times}S^1}$ of all nonempty compact subsets of $X{\times}S^1$ has the S-ECIP.

• The Journal of Natural Sciences
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• v.10 no.1
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• pp.13-16
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• 1998

Every complex vector bundle over $S^1$ splits sum of line bundle and the first Chern class classify complex line bundle. This implies every complex vector bundle over $S^1$ is trivial. In this paper, we show the existence of some nontrivial complex vector bundle over $S^1$ in the equivariant case.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.3
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• pp.569-581
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• 2011

Let a finite group R act smoothly on a closed manifold M. We assume that R acts freely on M except a union of closed submanifolds with codimension at least two. Then, we show that there exists an isomorphism between equivariant topological complex vector bundles over M and nonequivariant topological complex vector orbibundles over the orbifold M/R. By using this, we can classify nonequivariant vector orbibundles over the orbifold especially when the manifold is two-sphere because we have classified equivariant topological complex vector bundles over two sphere under a compact Lie group (not necessarily effective) action in [6]. This classification of orbibundles conversely explains for one of two exceptional cases of [6].

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.963-986
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• 2022

Given a dimension function 𝜔, we introduce the notion of an 𝜔-vector weighted digraph and an 𝜔-equivalence between them. Then we establish a bijection between the weakly (ℤ/2)ⁿ-equivariant homeomorphism classes of small covers over a product of simplices ∆^𝜔(1) × ⋯ × ∆^𝜔(m) and the set of 𝜔-equivalence classes of 𝜔-vector weighted digraphs with m-labeled vertices, where n is the sum of the dimensions of the simplicies. Using this bijection, we obtain a formula for the number of weakly (ℤ/2)ⁿ-equivariant homeomorphism classes of small covers over a product of three simplices.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.179-188
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• 2007

We prove the equivariant version of the semialgebraic local-triviality of semialgebraic maps.

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.253-267
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• 2015

Given a complex submanifoldM of the projective space $\mathbb{P}$(T), the hyperplane system R on M characterizes the projective embedding of M into $\mathbb{P}$(T) in the following sense: for any two nondegenerate complex submanifolds $M{\subset}\mathbb{P}$(T) and $M^{\prime}{\subset}\mathbb{P}$(T'), there is a projective linear transformation that sends an open subset of M onto an open subset of M' if and only if (M,R) is locally equivalent to (M', R'). Se-ashi developed a theory for the differential invariants of these types of systems of linear differential equations. In particular, the theory applies to systems of linear differential equations that have symbols equivalent to the hyperplane systems on nondegenerate equivariant embeddings of compact Hermitian symmetric spaces. In this paper, we extend this result to hyperplane systems on nondegenerate equivariant embeddings of homogeneous spaces of the first kind.

• The Journal of Natural Sciences
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• v.11 no.1
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• pp.11-14
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• 1999

Let G be a compact Lie group and let $\rho$ : G $\rightarrow O(2) be a homomorphism. Denote by V the G-module associated with $\rho$ and by S(V) the unit circle of V. In this paper, we show that if G is abelian, then a real G-vector bundle over S(V) is isomorphic to Whitney sum of real G-line or G-plane bundles.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.2
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• pp.157-161
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• 2007

We show that there is an equivariant symplectic embedding of a two-torus with a nontrivial action into a symplectic manifold with a symplectic circle action if and only if the circle action on the manifold is non-Hamiltonian. This is a new equivalent condition for non-Hamiltonian action and gives us a new insight to solve the famous conjecture by Frankel and McDuff.