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-Youp
*researcher*; 서동엽

- 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

- Appears in Collection
- MA-Theses_Ph.D.(박사논문)

- Files in This Item
- There are no files associated with this item.

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