• Kyungpook Mathematical Journal
• /
• 제46권3호
• /
• pp.399-408
• /
• 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.

• 대한수학회지
• /
• 제50권4호
• /
• pp.865-877
• /
• 2013

The disjoint union of mapping class groups of surfaces forms a braided monoidal category $\mathcal{M}$, as the disjoint union of the braid groups $\mathcal{B}$ does. We give a concrete and geometric meaning of the braidings ${\beta}_{r,s}$ in $\mathcal{M}$. Moreover, we find a set of elements in the mapping class groups which correspond to the standard generators of the braid groups. Using this, we can define an obvious map ${\phi}\;:\;B_g{\rightarrow}{\Gamma}_{g,1}$. We show that this map ${\phi}$ is injective and nongeometric in the sense of Wajnryb. Since this map extends to a braided monoidal functor ${\Phi}\;:\;\mathcal{B}{\rightarrow}\mathcal{M}$, the integral homology homomorphism induced by ${\phi}$ is trivial in the stable range.

• 대한수학회논문집
• /
• 제11권3호
• /
• pp.815-823
• /
• 1996

M. Kontsevich posed a problem on mapping class groups of 3-manifold that is if M is a compact 3-manifold with nonempty boundary, then BDiff (M rel $\partial$ M) has the homotopy type of a finite complex. Here, Diff (M rel $\partial$ M) is the group of diffeomorphisms of M which restrict to the identity on $\partial$ M, and BDiff (M rel $\partial$ M) is its classifying space. In this paper we resolve the problem affirmatively in the case when M is a Haken 3-manifold.

• 대한수학회보
• /
• 제50권4호
• /
• pp.1297-1302
• /
• 2013

We prove that the mapping class group of a non-Haken orientable irreducible 3-manifold is finite and we show that the quotient group of the mapping class group by the rotation group is virtually torsion-free if the manifold does not have 2-sphere boundary components.

• Korean Journal of Mathematics
• /
• 제19권4호
• /
• pp.409-421
• /
• 2011

We study a natural map from the braid group to the mapping class group which is called Harer map. It is rather new and different from the classical map which was studied in 1980's by F. Cohen, J. Harer et al. We show that this map is homologically trivial for most coefficients by using the fact that this map factors through the symmetric group.

• 대한수학회지
• /
• 제37권3호
• /
• pp.491-502
• /
• 2000

The disjoint union of mapping class groups g,1 gives us a braided monoidal category so that it gives rise to a double loop space structure. We show that there exists a natural twist in this category, so it gives us a ribbon category. We show that there exists a natural twist in this category, so it gives us a ribbon category. We explicitly express this structure by showing how the twist acts on the fundamental group of the surface Sg,l. We also make an explicit description of this structure in terms of the standard Dehn twists, as well as in terms of Wajnryb's Dehn twists. We show that the inverse of the twist g for the genus g equals the Dehn twist along the fixed boundary of the surface Sg,l.

• 한국과학교육학회지
• /
• 제19권3호
• /
• pp.479-486
• /
• 1999

중학교 생물의 광합성과 호흡단원에 대해 개인별 또는 협동 학습개념도를 적용한 수업결과를 전통적 강의식 수업결과와 비교하였다. 그 결과 개념도를 적용한 학습이 전통적 강의식 수업보다 효과적이었다. 공분산분석을 실시하여 개인별 개념도 작성그룹과 협동 개념도 작성그룹간(p<.008), 협동개념도 작성그룹과 전통적 강의식 학습그룹간(p<.0001)에 유의성있는 결과를 얻었다. 또한 협동개념도작성그룹이 개인별 개념도 작성그룹보다 효과적이었다(p<.04). 따라서 협동개념도 학습방법이 생물학습에 있어 가장 효과적이었다.

• 영어어문교육
• /
• 제7권1호
• /
• pp.137-160
• /
• 2001

This paper is to suggest the way to improve EFL reading skill through semantic mapping by leveled group activities. Semantic mapping is a categorical structuring of information in graphic forms or diagrams. It can be used to activate and organize background knowledge on topics in classrooms. For small group activities, the class is divided into higher leveled groups and lower leveled groups of four members based on their grades. The teaching process has three stages: Pre-reading, while-reading, and post-reading. In the pre-reading stage, students discuss what they know about the topic. They map ideas with a brainstorming technique. In the while-reading stage, they read the text about the topic. While they are reading, they could ask some questions they might have and discuss the information in the text and categorize them with semantic mapping. In the post-reading stage, they discuss what they thought of the topic and add some information about the topic with semantic mapping. For the subjects of this study, third grade, middle school students were selected: 41 students for the experimental group and 35 students for the control group. The experimental period covered almost one semester from March to August, 2000. The results were as follows: 1) The students in the experimental group had higher scores in reading comprehension than those in the control group when semantic mapping was used; 2) The use of semantic mapping in reading comprehension was found to be much more effective in the higher leveled group than in the lower leveled group; 3) The results of questionnaires showed that many students became more interested and motivated in English, and semantic mapping helped them to participate positively in reading the English text. Thus, using semantic mapping by leveled group activities can be an effective alternative to traditional teaching methods for teachers who desire to improve reading skill in middle school students' English classes.

• 한국과학교육학회지
• /
• 제29권8호
• /
• pp.898-909
• /
• 2009

이 연구에서는 화학 개념학습에서 역할놀이 비유가 대응 관계 이해도 및 대응 오류에 미치는 영향을 비유추론 능력에 따라 조사하였다. 중학교 1학년 학생 151명을 비교 집단과 '원 안에서 달리기' 역할놀이 비유를 사용한 처치 집단으로 배치하였다. 목표 개념인 '기체의 부피와 압력의 관계'에 대한 각 집단별 수업을 진행한 후, 바로 다음 차시에 대응 관계 이해도 검사를 실시하였고, 4주 후에 파지 검사를 실시하였다. 또한, 대표적인 대응 오류를 범한 학생들의 대응 과정을 조사하기 위해 면담을 실시하였다. 연구 결과, 역할 놀이 비유는 학생들의 비유 추론 능력에 관계없이 대응 관계 이해 및 대응 관계 이해 파지를 향상시키는 것으로 나타났다. 대응 오류에서도 역할놀이 비유를 사용한 처치 집단 학생들이 비교 집단 학생들보다 대응 오류의 빈도가 낮았다. 그러나 두 집단의 학생들 모두 유사한 대응 오류를 범하는 것으로 나타났고, 학생들의 비유 추론 능력에 따른 대응 오류 유형별 빈도 차이는 없었다. 이에 대한 교육학적 함의를 논의하였다.

• 대한수학회지
• /
• 제57권5호
• /
• pp.1103-1118
• /
• 2020

In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These multiarc graphs naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certain complexity conditions, that simplicial maps between them only arise in the "obvious way". We also observe that, again under necessary complexity conditions, subsurface strata are convex. Put together, these results imply that certain simplicial maps always give rise to convex images.