Topological properties of semialgebraic $G$-Sets = 준 대수적 $G$-집합의 위상적 특성에 관한 연구

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The topological properties of semialgebraic actions of semialgebraic groups on semialgebraic sets are studied. Let $G$ be a compact semialgebraic group. We prove that every semialgebraic $G$-set with finitely many orbit types has a semialgebraic $G$-$\CW$ complex structure. Using this result, we also prove that every semialgebraic $G$-set with finitely many orbit types admits a semialgebraic $G$-embedding into some semialgebraic orthogonal representation space of $G$ for $G$ a compact semialgebraic linear group. An affine semialgebraic $G$-set means a semialgebraic $G$-set which is semialgebraically $G$-homeomorphic to a $G$-invariant semialgebraic set in some semialgebraic representation space of $G$. Let $M$ and $N$ be affine semialgebraic $G$-sets. We find a one to one correspondence between the set of semialgebraic $G$-homotopy classes of semialgebraic $G$-maps from $M$ to $N$ and that of topological $G$-homotopy classes of continuous $G$-maps from $M$ to $N$. We also deal with the equivariant semialgebraic version of a theorem of J. H. C. Whitehead. We also deal with semialgebraic $G$-vector bundles. It is proved that any semialgebraic $G$-vector bundle over an affine semialgebraic $G$-set has a semialgebraic classifying $G$-map. Moreover, we prove that the set of semialgebraic $G$-isomorphism classes of semialgebraic $G$-vector bundles over an affine semialgebraic $G$-set $M$ corresponds bijectively to the set of topological $G$-isomorphism classes of topological $G$-vector bundles over $M$. Finally, we construct the equivariant Whitehead group of affine semialgebraic $G$-sets. It is shown that there is a well-defined Whitehead torsion for any $G$-homotopy equivalence between affine semialgebraic $G$-sets. We also prove the semialgebraic invariance of the Whitehead torsion. Moreover, we construct the restricti...
Advisors
Suh, Dong-Youpresearcher서동엽
Description
한국과학기술원 : 수학전공,
Publisher
한국과학기술원
Issue Date
2001
Identifier
166359/325007 / 000935138
Language
eng
Description

학위논문(박사) - 한국과학기술원 : 수학전공, 2001.2, [ vii, 106 p. ; ]

Keywords

transformation group theory; 벡터 번들; 호모토피; 준 대수적 집합; 변환군론; semialgebraic set; homotopy; vector bundle

URI
http://hdl.handle.net/10203/197750
Link
http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=166359&flag=dissertation
Appears in Collection
MA-Theses_Ph.D.(박사논문)
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