2006 ( 3 4 206 3 4 410) 1. K. Janich, Topology, Springer-Verlag, 1984; 2. M.A. Armstrong, Basic Topology, Springer-Verlag, 1983. 1. J.M. Lee, Introduction to Topological Manifolds, Springer, 2000; 2. J. R. Munkres, Topology, (Second Edition), China Machine Press, 2004; 3. J. Matousek, Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry, Springer, 2003 4. M. Henle, A Combinatorial Introduction to Topol- ogy, W.H. Freeman and Company, 1979. I (2 ) 2/20 Gauss Johann Benedikt Listing 1847 topology http://www-history.mcs.st-and.ac.uk/Mathematicians/Listing.html Georg Friedrich Bernhard Riemann Listing Riemann Surface http://zyx.org/Riemann.html Cantor 1870 Fourier Furstenberg Euler Euler stereometry 1750 1751 Euler Descarts Euler 1895 Poincare ‘ ’(Analysis Situs) Poincare Euler Von Staut Euler Thursten charge discharge Euler Moore Lefschetz Steenrod Wilde Steendrod Euler Sylvester-Gallai Euler Gauss Bonnet 1
Transcript
2006
( 3 4 ! " 206 #!$ 3 4 ! " 410)
%&1. K. Janich, Topology, Springer-Verlag, 1984; 2. M.A. Armstrong,
Basic Topology, Springer-Verlag, 1983.')()%*&1. J.M. Lee, Introduction to Topological Manifolds, Springer, 2000; 2. J.
R. Munkres, Topology, (Second Edition), China Machine Press, 2004; 3. J. Matousek,
Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics
and Geometry, Springer, 2003 # 4. M. Henle, A Combinatorial Introduction to Topol-
ogy, W.H. Freeman and Company, 1979.
I +, (2-
).2/20 /
Gauss 012 Johann Benedikt Listing 3 1847 456789: topology ;<=>?http://www-history.mcs.st-and.ac.uk/Mathematicians/Listing.html
Georg Friedrich Bernhard Riemann @ Listing ACBC1EDGFHI0JKMLNOPRiemann Surface 0QRST ? http://zyx.org/Riemann.htmlUCV HCICXWY .[Z\ /]HI_^`HI . ^`abcda0efghi/j_kl HI .]mn!op qrst/
Cantor 3 1870 4e Fourier u \ 0vwxy P e UV HI0vw ?z p| 8 V 0~L;~0 Furstenberg 0t \ ?Euler 7b0 & Euler 8N stereometry
G. C. Lichtenberg: What is the good of drawing conclusions from experience? I
don’t deny we sometimes draw the right conclusions, but don’t we just as often draw
the wrong ones?
(3/6)
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$%1-.+, |K~ P 1 32 1 2 v 34 1 4 +, 3e7 1 7 vk
Lawrence Narici, Partners: Functional Analysis and Topology, Topology Atlas
Invited Contributions, Vol. 6, 2001, available from: http://at.yorku.ca/t/a/i/c/
42.htm
$% ? 1 ? E$% 23 $% 2 #2 v n$% ? 1 2 ? l U $% 23 4
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4
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Mapping cylinder, Mapping cone, ^ Q_`aH ^bcd (Mobiuse)`agf JKhi EFbcdjk MNl K H im Fbcdjk Klein n H Klein
nopq mr Mobiusets
uwvwx OyP#zw Rw|~~~~~( www zw Rw~~~~~ ) ww
Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002 Chapter 0.
Some Underlying Geometric Notions.
6
Spring and All – William Carlos Williams
Now the grass, tomorrow / the stiff curl of wildcarrot leaf / One by one objects
are defined – / It quickens: clarity, outline of leaf
But now the stark dignity of/ entrance – Still, the profound change / has come
upon them: rooted they / grip down and begin to awaken
III f 4 s(3/22)
n Z EF R v n Z O:P R HausdorffO:P ! Z " q#$%'&( Z " q*)# K+#$ L )# KR,- op./ ( 0123 )
45 678 q:T 78 q9:;<=>#$ 78 q S178 q R
Riemann # KTopology may have had its tentative beginnings in isolated thoughts of Descartes,
Leibniz, and Euler, but it was Riemann who brought the subject into the mainstream of
mathematics with his inaugural dissertation in Gottingen in 1851. His introduction of
the Riemann surface in that year showed the indispensable role of topology in questions
of analysis, and thus ensured the future cultivation of the subject by the mathematical
community, if only for the service of analysis. In fact, of course, Riemann surfaces
were quickly seen to be of interest in themselves, and were the source of two ideas of
profound significance in later topology – connectivity and covering spaces. – John C.
Stillwell, Classical Topology and Combinatorial Group Theory, Springer, 1980.?@ )# KABC > Dw[&\w] M .E GF >'HGI |& GF RGJ'KL >NMPO LQRSR O:P ; u >UT Q_V ;WX YZ[\]^_R`badc !feMobius
e(crosscap)=
LP 2 g f fF =
`badhfifjfkl : j h`manc !e To (cap)=L
S2 g F =`manhijk : j h` !e anp ih Q_
=` m e anp ih !eqr
=L Q K g F
crosscap+handle= 3 crosscaps
fs @ Ivo Nikolov and Alexandru I. Suciu, RECOGNIZING SURFACES, http:
//mystic.math.neu.edu/inikolov/Topology/
It’s interesting to attempt to understand/prove that relation ”intuitively” - what
it says is that attaching a crosscap to a torus is indistinguishable from attaching one to
a Klein bottle. The ”visual” proof of this: start with the 2-sphere, attach a handle to
get a torus and then replace a disk with a Moebius strip to end up with T 2#P 2. Now
7
slide one end of the handle along the surface until you reach the embedded Moebius
strip, at which point you slide the handle’s end once around the Moebius strip. At
the end of that motion, push the end of the tube back into the complement of the
strip. When you’ve finished, you discover that the handle is now attached ”the other
way” – there are really only two ways to attach a handle to an S2, one of them giving
a T 2 and the other a Klein bottle, and the presence of the imbedded Moebius strip
in this surface gives you a means of swapping between the two means of attachment.
And, once you managed this feat, you’ve shown [T 2#P 2] = [K2#P 2]. (If anyone
knows a proof that doesn’t come off sounding like mysticism, I’d *love* to see it :-)
http://www.math.niu.edu/~rusin/known-math/01_incoming/conn_sum Hf/ )f# K F `a Vq T 2F
P 2^ V R > S2 q R T 2#P 2 = P 2#P 2#P 2 q ^ V u R b
)# RK P = 〈S | W1, · · · ,Wn〉, S q !e > Wi q Sh q 2R! >#"$ e !% a ∈ S > a
fQ!'& Y L( ) a−1fQ
!& Y L( F q 2.
&f( ) f q x %+*,3 K-K. )# f0/12 332 John Lee, Introduction to Topological Manifolds, Springer, 2000 Proposition
6.4.sKR45 X (folding/unfolding, cutting/pasting)6 i x ] M R7 ,*3)#8 KR45 X 9 Y 5:K; F R<<ffK = > 6 i 45 X , 3 K#K = K#T, P#K = P#T
Terry Lawson > Topology
A Geometric Approach > Oxford University
Press, 2003 > The Classification of Surfaces+@? > RABCD&FEG >HIJKL M +
w John C. Stillwell, Classical Topology and Combinatorial Group Theory,
Springer, 1980!FN Chapter 1. Complex Analysis and Surface Topology.
? N68 O9 Y crosscap+handle= 3 crosscaps
RPQR HR ; John H. Conway "#pS Q 9 Y R jk 2*3 (Conway’s ZIP proof)—J American Mathematical Monthly 1999 TkU uV >> 78 I &WN RAppendix C: Jeffrey R. Weeks, The Shape of Space, (Second Edition) Marcel
)f# pSf.f/ R !e;< 23P> 2 B. Mohar, C. Thomassen, Graphs on Sur-
faces, The Johns Hopkins University Press, 2001!N Chapter 3 Surfaces. =2 C.
Thomassen, The Jordan-Schonflies Theorem and the classification of surfaces, Amer.
Math. Monthly, 99 (1992), 116–130.
Graphs and surfaces form a natural link between discrete and continuous mathe-
matics. They enable us to understand both graphs and surfaces better. – B. Mohar,
C. Thomassen, Graphs on Surfaces, The Johns Hopkins University Press, 2001.
A surface is a two-dimensional manifold. The classification of compact surfaces
was “known”, in some sense, by the end of the nineteenth century, Mobius and Jordan
offered proofs (for orientable surfaces in R3) in the 1860’s. Mobius’ paper is quite
interesting; in fact he used a Morse-theoretic approach similar to the one presented in
this chapter. The main interest in Jordan’s attempt is in showing that the work of an
outstanding mathematician can appear nonsensical a century later. – M.W. Hirsch,
Differential Topology, p. 188, Springer, 1976.
If you are interested in research work on surfaces done by Gauss, read Chapter
1 (“Invitation to Topology”) in “A Mathematical Gift, I”, Amer. Math. Soc., 2003.
10
– Koji Shiga, Toshikazu Sunada, A Mathematical Gift, III: The interplay between
topology, functions, geometry, and algebra, p. 25, Amer. Math. Soc., 20057. ! d e5 V.I. ArnoldR V > On teaching mathematics
+ g q e R<e ^ > J ? k R 1 V http://pauli.uni-muenster.de/ mun-
steg/arnold.html
The theorem of classification of surfaces is a top-class mathematical achievement,
comparable with the discovery of America or X-rays. This is a genuine discovery of
mathematical natural science and it is even difficult to say whether the fact itself is more
attributable to physics or to mathematics. In its significance for both the applications
and the development of correct Weltanschauung it by far surpasses such ”achievements”
of mathematics as the proof of Fermat’s last theorem or the proof of the fact that any
sufficiently large whole number can be represented as a sum of three prime numbers.
IV f 20 sf3/29
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10
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’f f .fEf> 5: v ! g V R 5: v $ WR./&\ ? W R ./ R p)0 ` =) q x 12 >1 F !2$&] > x V >7 354565758595:<;>=@? 3@A@B@C@D5C@E@FHG5I@J@K@FML7/@N@O@P5Q@R@S@T@U5V@W@X@Y@Z8 [5\5] ;5L5^ ]5_ F 4 071 ]@_5`9 a@:@b@c@d@[@\@e@f FMg@h@D@i@j@R7; [@\ R@FMk 3@d Q@l@D@m f@n@[@\@`Mo@p@a@q@r@s@tvu@w@c@xy D5z5 t5u L5|5@~@@5@D@z@ t5u@`M3@c@5@@@5 FM@@5@@@Y@D ‘ [5\ + r5s5p55[5\
Linear Analysis: An Introductory Course, p. 95, Cambridge University Press, Second Edition, 1999.20 n;o;p M;E5¯ R K5¦5²;Líú5µ5¶;S;q;r ú?s; « I;t?uíú Banach µ5¶;T 8;v;w ;x «;y;z?% R K Weierstrass Approximation Theorem ;|;
y A)*)+ ¶è (éê "$ 2¸´ %ëì *+ E¸y A(íîïðñò § 2ó)ô)õ ;mö÷ 2a´ %";ø 'ùúàgh2aùêûgh2aü ýþÿ 2 þÿ 2 ÿ 2 § "ÿE H2 y A "à; 2 y A F " ; EÌ 5 "; ( E F Lé "à;øcEy A l ")$ ô vEì ô!"#$ 2% o E!
('&)(*+,-. 2
221/)E
It is hoped that this enterprise will make mathematics more widely used where
it is needed, and more accessible in fields in which it can be applied but where it has
not yet penetrated because of insufficient information. – Gian-Carlo Rota, Editor’s
Statement for Encyclopedia of Mathematics and its Applications.
(4/5)
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-0Ì1⇒Lindelof
Lindelof 23 ó ë 5 -0Ì1
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14
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OÀ( @ ,A,B,f(x) =
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5 A§
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B§
11E
(4/10)1 ô 2 ó ë457B1457 Þ145Þ4 E The Urysohn Lemma24
Janich L
The Urysohn Lemma
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2 ' 2)(* %+, 2 - À./01 step domain+2 #
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9u+I ¦ %KJ< ó ëL[,MN Ä , 2 MN ( Ì1,E
26
T4
ó ë# 7O>O>OP 7 9 u(Shrinking Lemma)
0T4
ó ëOQ 7O>O3O4O5U1, . . . , Uk
2 Ì ú %R 345V1, . . . , Vk
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27 T4
ó ëm 7>>P 79uThe Urysohn Lemma
Lm7>TU 4 ê VW 79u
28
Munkres, Topology, § 36 7X>XTXU 4 ê Ì L YZ ~\[)3]_^ ó ëm[ ó ë 2a`bXcMunkres,
One of the central aims of homotopy theory is the classification of continuous
functions. This is greatly facilitated if one has available either the homotopy lifting
property or the homotopy extension property. The homotopy lifting property is enjoyed
by fibrations, such as covering projections. The homotopy extension property is enjoyed
by cofibrations. – I.M. James, General Topology and Homotopy Theory, p. 2, Springer,
1984.
Rather than the arduous and systematic study of every new concept definable
with a graph, the main task for mathematician is to eliminate the often arbitrary and
cumbersome definitions, keeping only the “deep” mathematical problems. Of course,
the deep problems may well be elusive; indeed, there have been many definitions (from
Dieudonne, among others) of what a deep problem is. In graph theory, it should relate
to a variety of other combinatorial structures and must therefore be connected with
many difficult practical problems. Among these will be problems that classical algebra
is not able to solve completely or that computer scientist would not attack by himself.
– Claude Berge47, Foreword to the first edition of M.C. Golumbic, Algorithmic Graph
Theory and Perfect Graphs, Second Edition, Elsevier, 2004.
(4/19)
Riesz $H Ø h µG¶ ± 1 2Ỽ µG¶ 483 µ
¶ R *+ »%8 P h´ W 3» ¼ µ§¶ ±Tychonoff ¹H ¸ P¡¢ 49: 1. Alaoglu’s Theorem: Banach
45 d aba !" «$# Munkres, p. 186, Exercise 11.463/1547Claude Berge (1926–2002), %&'()a+*,- d./0$1 $2 l '$( s$34 5$%$67$89g+:g,- d;aiaj ($7$<# Topological Spaces : Including a Treatment of Multi-Valued Functions, Vector
Spaces and Convexity, Oliver and Boyd 1963 (republished by Dover Publications in 1997.)48 =ak> @?ABCg p. 21 @DE@FGHIJa 2001 K491. I.M. Singer, J.A. Thorpe %aMLNOPa iaj ( sQR ( mSTUVMW$X 8($YZHI$J V 1985 V
1.6 [$\ 2. Klaus Janich, Topology, Chapter X; 3. B. Bollobas, Linear Analysis: An Introductory
20
2. ! "$#%&'()*
3.'()*+, &-
A theorem that goes against intuition has its existence justified by this fact alone.
All right. An equally general, but perhaps more weighty point of view is that every
discipline must strive to clarify its own fundamental concepts. The concepts didn’t
come out and advertise themselves; it is the mathematician’s task to pick the most
convenient among several similar concepts; and the theorem of Tychonoff, for instance,
has been a decisive reason to give preference to the compactness concept defined via
open covers, over the concept of sequential compactness, which is not transferred to
infinite products. – Klaus Janich, Topology, Chapter X, §2 What Is It Good For.
Although many proofs in point-set topology sort of work by themselves, guided
by intuition and oiled by a cunning terminology and spacial intuition – the proof of the
Tychonoff theorem is not one of these. – Klaus Janich, Topology, p. 162.
Tychonoff ./01 2341. Henri Cartan 56 78 (filter)
9:;<>=Zorn’s Lemma ? 3@A78BCD AE78 (ultrafilter);
%FX&E78CGHC
X8%
C IJK%X \ C L AMN
2. L EO78OPQO 0OR TSOUVO L “ WX ⇔
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YZ"$ [ \ “@ AE78BPQ ⇔
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3. ] X A8^_FJMN`Oa XbcdBefg
ih 8icid"kj X&ia Ei7i8lPQ 50: m 2n3poiqrsD “
@ AiEi78BPQ⇒
”23" t :uvw xyz
4. ./ =|~8^_FJMN`a XbcdBefg h 8cd 51.
Filters were introduced by Henri Cartan (born july 8, 1904) in 1937 and sub-
sequently used by Bourbaki in their book Topologie Generale. An equivalent no-
tion called net was developed in 1922 by E. H. Moore and H. L. Smith. http:
//planetmath.org/encyclopedia/Filter.html
The universe is full of magical things, patiently waiting for our wits to grow
sharper. - Eden Philpotts
Course, pp. 118–119, Cambridge University Press, Second Edition, 1999; 4. =>V ?ABC V p. 96,
52 acbcdcecfcgcgchcicacb \ dcacbÃcjck UEl¹cmcnEoVqp crcs ¹ctcncu rEs ¥cvco ‘ n ´ ’hcicacb 53 w Ç ucx Æ ·Ey¢¹EzEE| ÇEE~ dEEEnEu¢®EEEE L u EEEEE¢Æ ® ÆE¢ w¢ÃEnEE Æc d Yc544/10
22
n=A * PQ ] uniform topology _ 55"L;
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compact-convergence topology (
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h8 "Te R -b
[! Arzela-Ascoli 01 " Munkres"
p. 290, Theorem 47.1, p. 293,
Exercise 4.
John L. Kelley, General Topology, Springer, 1955.
(4/26) ! C ih!" hI ^s 1" ! u i"IH6 h! " e ( ih ⇒ ih ), "
compactly generated, σ-
h bcd 57 I cd!" AinAC " $% J&a bcdB h !"bcd " i I#$ C Lindelof+
@ x B A43%( ⇒ 58;SY "'& () e** 1 Hausdorff
C "+ ! C Hausdorff
C-,-. 59: Janich, p. 125, Lemma (1);
Munkres,
Theorem 41.1
& h8!01 (shrinking lemma)60 : Munkres, Lemma 41.6
HausdorffC ;GH;J&a bcd V/D0 1 61:
Munkres, Theorem 41.7
X324YZ .56 1 x 7 + Stone-Cech7 " Stone-Cech
7
Ramsey 1 A89R Bela Bollobas, Modern Graph Theory, Theorem
32, p. 207, World Publishing Corporation, 2003.
Partitions of unity are most often used in mathematics to “patch together” func-
tions that are defined locally so as to obtain a function that is defined globally. – J.R.
Munkres, Topology, p. 259, Second Edition, China Machine Press, 2004.
First, you shouldn’t get the impression we’re always in the “local situation” just
because we have a chart before. ... Secondly: The construction of global objects
55Munkres, p. 266564/1757 hci ;: 6= < o ;: Ë58 = y?> ¥ u ´ K1 , K2, K3, · · · @ ¯ °?A?B V n u A;B J1 , J2, · · · , C?D Ki ⊆ Ji,E j > i + 2
Ji ∩ Jj = ∅.
59 F;G;H 4/5 I;J;K ´ Hausdorff ¯°Ã;L;M¯° \ ·cyON¢¸ OP n ´¢60 Q;R;SVUT;V ;W;X V 1.5.1 VUY;Z;[;\;]V 2005 ^ ·cy;_;`cf 4/10 T4
¯° Æ ;:;:;a µ61 b 4/10
;:;c;d ;e;f |;gOh ÆOi ¸ Ë23
starting from local data is certainly the main objective of partitions of unity. But they
can also be used to break up existing global objects into local ones and thus make
computations practicable. ... Thirdly and finally, let us mention that partitions of
unity don’t exist just for the sake of functions and sections in vector bundles, but fulfil
many other subtler ends. K. Janich, Topology, pp. 122 – 123.
V ] 2 ^_(5/8)[
(4/17):
s 0R " s COr . 62;
sr 6 -
b$'O( 7 $ s A7U>' 7 ; 0OR (O OHausdorff
"Os AU' ? sr "
Loop space Ω(X,x0): i V (S1, ∗),
(X,x0)i'i( ^ π1(X,x0):
Ω(X,x0)/ '^ L MN D loop space'
f ' g, f ′ ' g′ ⇒ f f ′ ' g g′ s$s ? 3sr .s Cs W Poincare C ?I 3-
¹W BÄ»²VW vA uÈ"» uv d u ¸¹ »Â_! ¯ _ uv 4 º»»Á É ¹ » _ v ¦g» ¯¢Ó ¦! m!n !ÉÑF!£» ¯ n W! T Ü#» ¯ #` d »æ f # « Ée¹Ü Men of Mathematics v! »|Ú ¢"# ¼ n !$!¡ d% !&Ð v ! ' Ç() ¹Ü*
Microbe Hunters + ,¤ & ± »ÑØ| ¢!¸p &-¼ m ¹! $¡BÄ» uv ± ¹×.¹×/X Ç ¸ ¹ Y 01!&-Ç3254^É)B_# µ6 v uv ± ÃÄ7aW»98: « ³´ ×; É n Ç TÜ#»=< « µ!uv Ó!¬ »ÙÚ>?@g!W Ç » Ú5AB fCD ¶ ÉFE µuv G uv#Ó!¬ º!»Ç)ÝHI»GJKLM» !$J ! » Z JG uv#Ó¬NÇ ÓOÇÉFPQRSJ4 vTU RFPQVWRFXY v3Z\[] R2004.
A friend of mine had cited a professor saying something like ”topology is like
psychology in trying to understand intrinsic properties of beings, and algebra is like
sociology in trying to understand the structure of their interactions with each other”.
Hey, I just realized this makes algebraic topology social psychology... Arkadas Ozakin
(5/10)^3_5` Ð( avbcdef )
... the student skit at Christmas contained a plaintive line: ”Give us Master’s
exams that our faculty can pass, or give us a faculty that can pass our Master’s exams.”
Paul Halmos, I want to be a Mathematician, (Washington 1985).
When one is truly interested in a specific question, there is usually very little in the
existing literature which is relevant. This means you are on your own. – M. Raussen,
C. Skau, I hope we shall continue–Interview with Jean-Pierre Serre, Nieuw Archief
voor Wiskunde, 5 (2004), 38 – 41. Available from: http://www.math.leidenuniv.
ogy studies complicated properties of simple spaces (manifolds, polyhedra) . – A sem-
inar announcement posted in Moscow University in 1950s
If we go over from a topological space to the simplex numbers and incidences
of a simplicial complex homeomorphic to that space, we don’t have yet any topolog-
ical invariants, but we can be sure that all topological invariants can in principle be
calculated from these data, because they are enough to recover the original space up
33
to homeomorphism. This observation is so to speak the starting point of algebraic
topology, and for decades all efforts were canalized in this direction indicated by it.
What eventually came out was, expressed in today’s terminology, the first significant
algebraic topological functor, namely simplicial homology. – Janich, p. 92.
The investigation of simplicial complexes led to the development of homology
theory, and homology theory itself can be used in the algebraization of attaching maps.
The homological properties of attaching maps (as I must say somewhat vaguely, since
getting into homology in more detail would lead us too far afield) can be expressed by
means of certain “incidence numbers”. Such numbers do not contain full information
about the attaching maps any more, nor do they allow us to recover fully the topology of
the complex. But they are enough to determine the homology of the complex. Janich,
p. 103.
“Simplicifying assumptions” make life easier for the mathematician, but when can
they be made? In algebraic topology a compromise often has to be worked out: The
spaces must be special enough to allow certain methods to work and certain theorems to
apply, but at the same time they must be general enough to include certain important
examples of applications. Considering CW-complexes (or spaces homotopy equivalent
to them) is often a good compromise of this sort, and this is another reason to be
acquainted with the notion of a CW-complex. – Klaus Janich, Topology, pp. 104 - 105.
Computing homology with simplicial chains is like computing integrals∫ b
af(x)dx
with approximating Riemann sums. – A. Dold, Lectures in Algebraic Topology, 1972,
p. 119.
Simplicial complexes or triangulations (first introduced by Poincare) provide an el-
egant, rigorous and convenient tool for studying topological invariants by combinatorial
methods. The algebraic topology itself evolved from studying triangulations of topo-
logical spaces. With the appearance of cellular (or CW) complexes algebraic topology
gradually replaced the combinatorial ones in topology. However, simplicial complexes
have always played a significant role in PL topology, discrete and combinatorial geom-
etry. The convex geometry provides an important class of sphere triangulations which
are the boundary complexes of simplicial polytopes. The emergence of computers re-
sulted in regaining the interest to “combinatorial topology”, since simplicial complexes
provide the most effective way to translate topological structures into machine lan-
guage. So, it seems to be the proper time for topologists to make use of remarkable
achievements in discrete and combinatorial geometry of the last decades, which we
34
started to review in the previous chapter. — Victor M. Buchstaber, Taras E. Panov,
Torus Actions and Their Applications in Topology and Combinatorics, p. 21, American
Mathematical Society, 2002.
A polyhedron is a topological space which admits a triangulation by a simplicial
complex. Thus we start with a study of the category of simplicial complexes. A sim-
plicial complex consists of an abstract scheme of vertices and simplexes (each simplex
being a finite set of vertices). Associated to such a simplicial complex is a topological
space built by piecing together convex cells with identifications prescribed by the ab-
stract scheme. Since the topological properties of these spaces are determined by the
abstract scheme, the study of simplicial complexes and polyhedra is often called com-
binatorial topology. – Edwin H. Spanier, Algebraic Topology, p. 107, Springer-Verlag,
1966.
(5/31), 4 @ ? 1 78
, @, A Z@ D Z@LM 56 ∂ =
0 ∂0 0 · · · 0
0 0 ∂1 · · · 0...
......
. . ....
0 0 0 · · · ∂n
0 0 0 · · · 0
∂2 =
0 0 ∂0∂1 0 · · · 0
0 0 ∂1∂2 · · · 0. . .
. . .. . .
...
0 0 ∂n−1∂n
0 0
0
= 0
6l D;? ! D 6 ! H l0m Ker(∂)/Im(∂) =
⊕iHi
Ker(exp(∂) − I)/Im(exp(∂) − I) =⊕
iHi79???
k "l8m # k + 1 $N78 %'&'( ∅ )c'*'+','-'.ca0/'1ji3204ca0506jr07'809jajr0:cu wj0; e'<0=jr0:ju79 > '? '@'A'B ajDCjje0E0/'?0Fji3G'H0F0I0J0K'L0Mja0+0,'-0.jjj
35
@
, " A ND@
, BC= f4BCD x =] 1'D = - $ % @ q U 1 >i D D a * l0mJ 1
l8m! R " ! Df#$%!& FlJ! R f )'K D 9 D f#$ K 9 & Fr() D l8m+*lJ, %-. D ,/ uv D ,/01Intuitively the “i-th homotopy group” describes the “i-dimensional round holes”
and the “i-th homology group” reveals the number of “i-dimensional rooms” in a geo-
metrical object. – Hajime Sato, Algebraic Topology: An Intuitive Approach, American
Mathematical Society, 1999.
The homology groups of a topological space are among the most complicated
constructions in all mathematics. To appreciate this, let us review the steps of the con-
struction. (1) Given a topological space, we must first divide the space into polygons,
edges, and vertices identified in some way – in a word, we must view the topological
space as a complex. (This is the geometric step of the construction.) (2) Given a
complex, we extract the incidence coefficients with which we define the chain groups
and the boundary operator. (This is the combinatorial step of the construction.) (3)
Finally, we isolate the subgroups of cycles and boundaries and define the homology
groups. (This is the algebraic step of the construction.) ... Thus a given topological
space has only one set of homology groups, and these groups are independent of the
manner in which the topological space is viewed as a complex. It follows that, whatever
information the homology groups contain, this information is of a purely topological
nature unaffected by the geometric, combinatorial, and algebraic steps of the construc-
tion. ... With invariance, we can be confident that the information carried by the
homology groups concerns the topological space alone, even though we have no idea
yet exactly what that information is! – Michael Henle, A Combinatorial Introduction
to Topology, pp. 153-154.
, & F D !& F H ! lJN;l0m l , lm 32 simplicial category h category of graded abelian groups 4 1BC , 4D 5 4D76 S & F,89 Df4'BIC lm =: ;< 4=> , l8m N f4BC? C 6 S & F@A8,l8m ? C l DCB DlJ & F@Al8
36
l 80f4BC % l8m 0 H $ Smith 81:
WA> l8 &(In homotopy theory it is clear at the outset that homeomorphic spaces have iso-
morphic homotopy groups, but the computation of these groups is, in general, inordi-
nately difficult. The situation is reversed in homology theory. Although the topological
invariance of the simplicial homology groups is quite a deep theorem, the computation
of these groups for a given space is accomplished by an almost mechanical procedure.
Clearly, this latter state of affairs possesses distinct advantages over the former in that
the difficult part of the theory (the proof of topological invariance) need only be done
once. In addition, the simplicial homology groups are all finitely generated and Abelian,
so their structure is rather easily discerned. Of course, the price we must pay for these
advantages is the rather severelly restricted class of spaces to which the theory applies,
that is, polyhedra. Although there are more general homology theories that embrace
a wider class of spaces (e.g., Cech homology theory and singular homology theory),
they will not concern us here. – Gregory L. Naber, Topological methods in Euclidean
spaces, p. 111, Cambridge University Press, 1980.
(6/5)( ) , l8m space→complex→oriented complex→groups of
chains →homology groupsR "l8m H 64 "l8m +N "l0m (
C D O )
With hindsight, one can say that homology theory bagan with the Desartes-Euler
polyhedron formula. It took a further step with Riemann’s definition of the connec-
tivity of a surface, and the generalization to higher-dimensional connectivities by Betti
1871. All these results have to do with the computation of numerical invariants of a
manifold by means of decomposition into “cells”; the computations involve only the
numbers of cells and the incidence relations between them, and it is shown that certain
numbers are independent of the particular cellular subdivision chosen.
However, these results remained isolated until they were forged into a theory of ho-
mology by Poincare 1895. Poincare felt that many barnches of mathematics were
80 "!811.J.R. Munkres, ( #%$%&%' ) (j%)%*%+%,%-/.%0ji21 wjx i §11, 3%4%.%509%6%7ji 1991. 2. 8%9: ilr':c~ci21 wcx i §5, ;+'967ci 2002.
37
clamouring for such a theory, but his immediate objective were to generalize a duality
relation observed by Betti, and to give a completely general version of the Euler for-
mula. The Betti numbers, as they have been called since Poincare introduced the term,
generalize the notion of connectivity number (genus) for an orientable surface. If n is
the maximum number of closed cuts which can be made in a surface without separating
it, then such a maximal system of curves a1, · · · , an constitutes a basis in the sense that
for any other curve p some “sum” of a1. · · · , an bounds a piece of surface in combi-
nation with p. The latter property of n, made precise, is meaningful for manifolds M
of possibly higher dimension than 2, and it serves to define the one-dimensional Betti
number B1. One immediately generalizes to the k-dimensional Betti number Bk when
curves are replaced by suitable k-dimensional submanifolds of M. Betti had observed
that B1 = B2 when the dimension m of M was 3. ...... Torsion is present when an
element a does not form a boundary taken once, but does when taken more than once.
An example is the curve a in the projective plane P which generates π1(P). Then a2 is
the boundary of a disc, though a itself does not separate P. Poincare justified the term
“torsion” by showing that (m−1)-dimensional torsion is present only in an m-manifold
which is nonorientable, and hence twisted onto itself in some sense.82 – John Stillwill,
Classical Topology and Combinatorial Group Theory, Springer, 1980. l8m 9 l0m N R m (acyclic): l8m % : Betti Euler-Poincare 83;Euler % lq U (pseudomanifold)84f 8 n " n "l8m ! q f 0 n "
n "l8m : 85# ! lm % ; q U 1 ?;A ]; f ; N f Betti 86 A ] 45 % q U9 ] Sn l8m 87
Sn Sm lo B o n = m
"l ( l )45 % q U 88
82 (c ! "+ # abel uca $ %cci'&'/ci'& (cuci'&'Ccca*) +j', -*. / 0j21 % 3 4*5 a7698j83 : ; 0 > ce'Ccja*<cuji>= zc*?*@ (j,*Ajj*8 B0+j ilr 2/2084Fred H. Croom, Basic Concepts of Algebraic Topology, p.3085Fred H. Croom, Basic Concepts of Algebraic Topology, p.3686 2c C w D E E D r0:cu87Sn @ /ca n D F G .ciIH Hn(Sn) = Z; Sn a J Kcr':cu E Bn+1 a w L iIMce K N 2jrja
n D O P88Rn − p E Sn−1 r Q R
38
Brouwer No Retraction Theorem:89 q `2 Bn+1 h Sn ' & 90
Brouwer Fixed Point Theorem: Bn+1, n ≥ −1,: r 6 & q 1
: r f ;< 4 "m 1Every finite hereditary set system can be regarded as an abstract simplicial com-
plex, and it specifies a topological space (the polyhedron of a geometric realization)
up to homeomorphism. Simplicial maps of simplicial complexes yield continuous maps
of the corresponding spaces. ... Conversely, if a topological space admits a triangu-
lation, it can be described purely combinatorially by an abstract simplicial complex.
(This description is not unique.) ... A continuous map, even between triangulated
spaces, generally cannot be described by a simplicial map. On the other hand, there
are theorems stating that under suitable conditions, a continuous map is homotopic to
a simplicial mao between sufficiently fine triangulations of the considered spaces, and
it can be approximated by such simplicial maps with any prescribed precision;... J.
Matousek, Using the Borsuk-Ulam Theorem, pp. 15–16, Springer, 2003.
−1.94/0 010102 035467@A 98 : ;< : . =0>?;@0A /BCD E .GF : f ,' HI
f J;< : .K = 95. LM : N + K DO89 PRQRSRT Stokes URVRWRXRYRZR[R\R]R^ T. W. Gamelin, R. E. Greene, Introduction to Topology,
pp. 168–169, CBS College Publishing, 1983.90 _a`abac f, d H(x, t) = f((1− t)x) WaX Sn eafagahaiajakalahai YamRnRoqpa[ Sn jarasat manu owvax0y91 za Y Hopf Classification Theorem |aa~aaaaay92n- RRY n- mRRR Z, Z Z YRmR f f(1) RRURwRmRRRRRRRRXaaa PaQ R ZR[R\R] J. Matousek, Using the Borsuk-Ulam Theorem: Lectures on Topological
i=1 Fi = Sn, E B+C i, R S Fi ∩−Fi 6= ∅: Sn ,++[+8+\]^\]`_ab[8c 6 d. x ∈ Fi *6M d(x, Fi) = 0. de f(x) = (d(x, F1), · · · , d(x, Fn)). L 4, Af: x, f(x) = f(−x).
Zf(x) g h 6 2 $E x ∈ Fn+1 ∩ −Fn+1;
Zf(x)
.ii
V h 6 jE x ∈ Fi ∩ −Fi.
5⇒2: Sn−1 Akl n+ 1 V Dm ;<E . WYgno Z BC 9 Sn : Sn−1
.; ;<E : g, E p WY C g q .rs J 5 tuv2⇒1: Sn Akwx Ny V Bn z| .~ Sn−1 x Bn
B :Z I+ C Sn−1 ; ;0<0E : EAk F, Sn : Sn−1
.+; ;<E :
(6/12)
Kneser KG2n+k,n: EY ( [2n+k]n
), AB ~ A ∩B = ∅
965/2497J. Matousek, Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics
and Geometry, Springer, 2003.98 6/5, Brouwer No Retraction Theorem99 Za[ 6/14 WaX
1003/29, 5/24
40
Kneser
Kneser KG2n+k,n
. E+ χ(KG2n+k,n) k+ 2. KneserC
1955 . χ(KG2n+k,n) ≤ k + 2.
L. Lovasz (1948 – ) M 1978 + `b χ(KG2n+k,n) ≥ k + 2,
9 Kneser O Kneser-Lovasz
+ +R S 9 x . V 101.C S Lovasz !" k" I. Barany 102
C## $&%' b V( O)*+ *! gR,- N Lyusternik-Schnirel’man Theorem*
Gale Lemma. Harvey Mudd College !./0 Joshua E. GreeneC
2002 'Lyusternik-Schnirel’man Theorem ! 5 O 721,L ` V34R Gale
Lemma ! U F1, · · · , Ft Sn
Bt V5+Y+ Ft+1 , · · · , Fn+1 Sn
Bn+ 1− t VXW-Y+ Z
∪n+1i=1 Fi = Sn, E B+C i, R S Fi∩−Fi 6= ∅ 103:
/6 d+e t = n+1 !7+7 104; U ε 89+c: \X] Sn !5+n+o F1, · · · , Fn+1 ! Lebesgue 0><; I VE x1, · · · , xm,
R S ∪mi=1B(xi,
ε2) = Sn; = Gi = ∪j: B(xj ,ε)⊆Fi
B(xj,ε2), E Gj ⊆ Fj , G1, · · · , Gn+1 F
x Sn !XW-n+o0>?>"; G1, · · · , Gn+1 Lyusternik-Schnirel’man Theorem k@x
Joshua E. Greene ; Kneser ! 105
A number of important results in combinatorics, discrete geometry, and theoretical
computer science have been proved by surprising applications of algebraic topology.
Lovasz’s striking proof of Kneser’s conjecture from 1978 is among the first and most
important examples, dealing with a problem about finite sets with no apparent relation
to topology. – J. Matousek, Using the Borsuk-Ulam Theorem: Lectures on Topological
Methods in Combinatorics and Geometry, Preface, Springer, 2003.
Algebraic topology provides measures for global qualitative features of geometric
and combinatorial objects that are stable under deformations, and relatively insensitive
to local details. This makes topology into a useful tool for understanding qualitative
geometric and combinatorial questions.
Considerable momentum has developed in recent years towards applications of alge-
braic topology in various contexts related to data analysis, object recognition, discrete
and computational geometry, combinatorics, algorithms, and distributed computing. –
101 A<B<C 2005 European Prize in Combinatorics D<E<F DMITRY FEICHTNER-KOZLOV G<H<Ia«J<K a<L Y<M<NR^ http://www.inf.ethz.ch/personal/dkozlov/publ.html102 O<P<Q<R<S<T Fa·VU L<W 30 X<Y<Za« P <[<\ QVR<]VT X V^<_ T XV`Va<bVcRYVdVea103 f<g £ Greene h<iaY Lyusternik-Schnirel’man Theorem Y P<j<k<l 104 ²a³<m<n <o s Y t U , Uo = x ∈ Sn : d(x,U) < 1 − diam(U)
q ) ! c $' ! c ' Q+2! #"%$& 7! "%'( *)+ ,- 9 . /0 1+2! ; 2 ! ) 3 ! c 106: 4 Sn w
x y V Bn z Sn−1 x M N Ak56 X Sn ! - 789 h R S: V+ : ! Si ; N+Z<78= !1+ U f V ; 2! $&>?++ A+k@ ) g 9 h ABk" BC t+T $&C 7D ; 2 $& f ! $E&>E? F+GEH+U f3JI!K N $E& q ) Q+ 0
106J. Matousek, Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics
and Geometry, pp. 43–44, Springer, 2003. Matousek fMLMNMOMPMQ fSRST c Y Z [MUSV ^ S.
Lefschetz, Introduction to Topology, Princeton, 1949. f P a Za[WUWVaa\YXYZ Armstrong Y T Y S[ 42
deg(f) +++ d+e+2 . a. ( ∂,f# J a# ,
x+a#(x) = 0 +++ x = y+a#(y) 107. U Si = A+i +A−
i , ∂A+i = ∂A−
i = Si−1 =
A+i−1 +A−
i−1. f#(A+i ) = X+
i , f#(A−i ) = X−
i ./ f#(Sn) = X+
n +X−n 6= 0 A
( C QE+2 . ). D E (I+a#)X+n = X+
n +X−n = 0, 9 X+
n = (I+a#)dn, BC dn ∈ Cn,
∂(X+n ) = ∂((I + a#)dn). (1)
HU dk ∈ Ck R S∂(X+
k ) = ∂((I + a#)dk), (2)
dk−1 ∈ Ck−1 R S
∂(X+k−1) = ∂((I + a#)dk−1). (3)
D (I + a#)(X+k−1) = Xk−1 = ∂(X+
k ); L Eq. (2), S : (I + a#)(X+k−1) =
∂((I+a#)dk) = (I+a#)(∂(dk)); - X+
k−1 +∂(dk) ∈ Ker(I+a#) = Im(I+a#),
9 Eq. (3) DQ BC d0 ∈ C0, R S
∂(X+0 ) = ∂((I + a#)d0). (4)
∂(X+
0 ) = ∅ 6= 0 = ∂((I + a#)d0),S : JK
- ! 789 h * tT $&>? w ' ! c ' A U H ′N \] H !^\] f : (H,H ′) → (H,H ′) f f ! H/H ′ D 3 jE tr(f) = tr(f |H′) + tr(f).
![Bibliography - Springer978-3-662-12918-0/1.pdf · Bibliography Adams, ... Brocker, Th., und K. Janich [I] Introduction to Differential Topology. Cambridge: Cambridge University Press,](https://static.documents.pub/doc/80x56/5b85e9807f8b9a2e3a8bcab3/bibliography-springer-978-3-662-12918-01pdf-bibliography-adams-brocker.jpg)
