• Kyungpook Mathematical Journal
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• 제61권1호
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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.

• 대한수학회보
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• 제40권1호
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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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• 제55권1호
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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.

• 자연과학논문집
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• 제10권1호
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• pp.13-16
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• 1998

원 위에서의 모든 복소 vector bundle은 line bundle로 나누어지며 첫째 Chern class는 복소 line bundle을 분류한다. 이것은 원위에서의 모든 복소 vector bundle은 trivial 하다는 것을 의미한다. 이 논문에서는 군작용이 있을 경우에는 원위에서의 복소 vector bundle중에 trivial하지 않는 bundle이 존재함을 보였다.

• 충청수학회지
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• 제24권3호
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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].

• 대한수학회지
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• 제59권5호
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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.

• 대한수학회지
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• 제44권1호
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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.

• 대한수학회논문집
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• 제30권3호
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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.

• 자연과학논문집
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• 제11권1호
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• pp.11-14
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• 1999

군 G가 compact Lie군이며 $\rho$ : G $\rightarrow$ O(2)가 homomorphism일 때 군 G가 가환군이면 원위에서 실 G-vector bundle은 실 G-line bundle들의 Whitney 합이거나 G-plan bundle들의 Whitney 합과 isomorphic 하다는 것을 보였다.

• 충청수학회지
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• 제20권2호
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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.