• 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.50 no.4
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• pp.865-877
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• 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.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.815-823
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• 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.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1297-1302
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• 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
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• v.19 no.4
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• pp.409-421
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• 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.

• Journal of the Korean Mathematical Society
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• v.37 no.3
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• pp.491-502
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• 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.

• Journal of The Korean Association For Science Education
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• v.19 no.3
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• pp.479-486
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• 1999

This study was to inquire the effectiveness of concept mapping on the learning of biology concept. In this study the achivement acquired by the students in the classes of individual concept mapping and cooperative concept mapping were compared with those of traditional lecture on "photosynthesis and plant respiration". The concept mapping teaching methods, by individuals and groups. were more effective than traditional one. The results of analysis of covariance (ANCOVA) showed that these difference were statistically significant; individual concept mapping was more effective than traditional class result(p<.008). Also the cooperative concept mapping was effective than traditional class result(p<.0001). And the cooperative concept mapping was more effective than the individual one(p<.04). So that cooperative concept mapping is the most effective method in biology class.

• English Language & Literature Teaching
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• v.7 no.1
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• pp.137-160
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• 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.

• Journal of The Korean Association For Science Education
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• v.29 no.8
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• pp.898-909
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• 2009

In this study, we investigated the influences of the role-playing analogy in chemistry concept learning on mapping understanding and mapping errors by analogical reasoning ability level. Seventh-graders (N=151) at a middle school were assigned to the comparison group and the experimental group. The students of the experimental group were taught with the 'running in the circle' role-playing analogy. After the students were taught about 'the relation between volume and pressure of gas', the test of mapping understanding in the next class and the retention test four weeks later were administered. The students with typical mapping errors were also interviewed to investigate their mapping processes. The results revealed that the role-playing analogy in chemistry concept learning improved mapping understanding and its retention regardless of analogical reasoning ability level. It was also found that the students in the experimental group had fewer mapping errors than those in the comparison group. However, there were similar patterns of mapping errors in both groups, and there were no significant differences in the frequencies of each type of mapping errors by analogical reasoning ability level. Educational implication of these findings are discussed.

• Journal of the Korean Mathematical Society
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• v.57 no.5
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• pp.1103-1118
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• 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.