??? ????? ????. ? ???? ?? ???? ???? ??? topology?? ??.

1
????

• ??? ?? ??
• ??? ??, ??
• ???, ?????
• ???? ??
• 3-?? ???
• ??

2
???????

• ??? ????? ????. ? ???? ?? ???? ???? ???
  topology?? ??.
• ? ?? ??? ????? ??. ???? ???
• ????? ??? ???? 2, 3, , n?? ??? ????? ???? ??.
  ?? ???? ?? ??? ????.

3
Topology? ??

• A set X along with a collection T of subsets of
  it is said to be a topology if the subsets in T
  obey the following properties
• 1. The (trivial) subsets X and the empty set
  are in T.
• 2. Whenever sets A and B are in T, then so is
  their intersection. (??)
• 3. Whenever two or more sets are in T, then so is
  their union (?? ??)

4
?

• ? , X
• ?, 1, 2, 1, 2????? P(X) A A? X?
  ????? topology??. Discrete topology?? ??.
• 1,2? ?? topology ?, 1,2,?, 1, 1, 2,
  ?, 2, 1,2, ? , 1, 2, 1,2

5
?

• ??? ????? ???? ?? ??(????)? ???? ??? topology
  ??. ?? ?? ?? ??? ???? ????. ?? ????? ????? ???
  ????.
• ????? ?? ???? ?? ???? ???

6
?? (2?? ???)

• ???? ????? 2-??? ????? ???.
• ?????? ??? ????? ????? ?? ?? ???.
• ?? ??? ?? ?? ??.
• ??? ??? ?? ?? ??. (? ???? ?)

7
?? ??? ?

• ??? ?? ???? ??? ?? ???. ?? ????? ???. ? ??? ?? ??
  3-?? ????? ???? ?? ??? ??? ?? ??.

8
?

• ?

• ?????

9
?

• ???

• ????

10
?

• ?? ????? ??????.

11
???(torus)
12
?????
13
http//web.meson.org/topology/projective.html
14
(No Transcript)
15
(No Transcript)
16
??? ?(Klein bottle)
17

• http//www.ma.umist.ac.uk/kd/geomview/geometry.htm
  l

18
(??)??? ??

• Connected sum construction.Remove two disks from
  two surfaces and identify the two circle
  boundaries.
• ?? ???? ???.
• ??(genus) ???? ???? ??, ?? ???? ???????
  ?????? ??, ?? ????(cross-cap)? ??
• http//www.math.ohio-state.edu/fiedorow/math655/c
  lassification.html

19
??? ???

• ???? ???? ???. ?? ???? ???? disk?? ??.
• ??? ??? ??? ???? ???? ?? ?? ??. (? ???)
• ???? ??? ??? ??? ??? ????? ????? ????
• ??, ??? ??? ??? ??? ????? ??.

20
??? ?? ??
2

• ?
• ???
• ??? ??
• ??? ?
• ?? ?? g
• ??? ?? g

0
1
0
2-2g
1-g
21
???, ???? ?

• ?? ??? ???? ??, ?? ??? ???? ???????

22

• ? ? class? ???? ?? ??. AB (A? ??? ??? B? ????
  ??)
• Loop?? ?? ?? ???.
• ??? ?????? ??.
• ?????? ?? ??? ???? ??? ????? ????.

http//mathworld.wolfram.com/Fundamental
Group.html
http//mathworld.wolfram.com/HomotopyGroup.html
23
??? ??

• n?? ???? ????? n?? ???? ??? ????? ??? ???. 2???
  ????? ???? ??? ??.
• ? S3 v? R4 v1, S1xS1xS1

http//geometrygames.org/
http//geometrygames.org/TorusGames/index.html
http//www.math.tu-berlin.de/diskregeom/stellar/
24
(No Transcript)
25
12?? 3-?? ???

• ? 12???? ????? ?????? 36? ??? ?? 1/10 360 36
• 3/10 360? ??? ??? ??.

http//www.nature.com/nature/journal/v425/n6958/fi
g_tab/nature01944_F3.html
http//www.geometrygames.org/CurvedSpaces/
26
???? ??(Poincare Conjecture)

• Poincare? ?? ??? ??? ???, ?????? ? trivial??? ?
  ??? ?? ???? ?????.

http//mathworld.wolfram.com/PoincareConjecture.ht
ml
27
??

• 2?? ?
• S. Smale 7?? ????
• Stallings, Zeeman 5?? ??
• Freedman 4??
• 3-?? ?? ??

28
Geometrization of 3-manifolds

• ?? 3-?? ???? essential??? ????? ???? 3-?? ?????
  ???.
• 3-?? ???? ?? ???? ? ???? ???? ??? ???? ???. ??
  ????? ??? ?? ??? ???.

http//mathworld.wolfram.com/ThurstonsGeometrizati
onConjecture.html
29
3-?? ???

• Euclidean geometry
• Hyperbolic geometry
• Spherical geometryhttp//en.wikipedia.org/wiki/Sp
  herical_geometry
• The geometry of S2 x R
• The geometry of H2 x R
• The geometry of SL2R
• Nil geometry, or
• Sol geometry.

30
??? ???? ???
?? ?? ??? ???? ??
http//www.nature.com/cgi-taf/DynaPage.taf?file/n
ature/journal/v425/n6958/abs/nature01944_fs.html