Takashi Tsuboi Graduate School of Mathematical Sciences, the University of Tokyo, Japan

Transcription

1 TOPOLOGY Takashi Tsuboi Graduate School of Mathematical Scieces, the Uiversity of Tokyo, Japa Keywords: eighborhood, ope sets, metric space, covergece, cotiuity, homeomorphism, homotopy type, compactess, coectedess, homotopy, homotopy group, simplicial comple, homology group, cohomology group, Euler-Poicaré characteristic umber, Poicaré duality theorem, Poicaré-Hopf theorem, Morse theory, Poicaré cojecture, itersectio umbers, likig umbers, kot theory Cotets Itroductio 2 Covergece of sequeces, cotiuity of maps, geeral topology 2 Metric Spaces ad the Covergece of Sequeces 22 Abstract Topology o Sets 23 Separatio Aioms ad Coutability Aioms 24 Compactess 3 Coectedess ad homotopy theory 3 Coectedess ad Homotopy 32 Homotopy Groups 33 Homotopy Eact Sequece 4 Simplicial complees ad homology theory 4 Simplicial Complees 42 Chai Complees ad Homology Groups 43 Sigular Simplicial Comple ad Sigular Homology Groups 44 Cochai Complees ad Cohomology Groups 5 Applicatios for maifold theory 5 Poicaré Duality Theorem 52 Poicaré-Hopf Theorem 53 Morse Theory 54 Homotopy Type ad Homeomorphism Type 55 Itersectio Numbers ad Likig Numbers 56 Kot Theory Glossary Bibliography Biographical Sketch Summary Topology is a brach of mathematics which studies the shape of spaces ad the cofiguratios of a space mapped i aother space The curret theory of topology cosists of geeral topology which is the foudatio to talk about covergece i various spaces ad cotiuity of maps betwee spaces, homotopy theory ad homology theory which defie various ivariats to distiguish spaces ad cofiguratios, ad applicatios of these geeral results to the iterestig objects such as maifolds Ecyclopedia of Life Support Systems (EOLSS)

2 Itroductio Topology is a brach of mathematics which studies the shape of spaces ad the cofiguratios of a space mapped i aother space Oe of the origis of topology theory is usually attributed to Euler s argumet to deduce the impossibility to fid a route i Köigsberg passig each of 7 bridges oce ad oly oce The cofiguratio of the bridges of Köigsberg is represeted by a graph with 4 vertices ad 7 edges, the questio is the eistece of a path passig through each edge oce ad oly oce Euler looked at the umber of attached edges at each verte, ad foud that for ay coected graph with at most 2 vertices with odd umber of attached edges, it is possible to fid such a path ad otherwise it is impossible to fid such a path I the graph represetig Köigsberg, the 4 vertices have 5, 3, 3 ad 3 attached edges, respectively, ad hece it is impossible to fid the desired route Euler also foud that for a cove polyhedro, ( the umber of vertices) ( the umber of edges) + ( the umber of faces) = 2 The umber of left-had-side does ot deped o the polyhedral arragemets ad it is called the Euler-Poicaré characteristic umber This is still the most importat ivariat to distiguish the shapes of polyhedra Aother origi of topology is the study of the covergeces of sequeces ad series such as Fourier series It took rather log time to uderstad that the poitwise limit of cotiuous fuctios may ot be a cotiuous fuctio while the uiform coverget limit of cotiuous fuctios is cotiuous After the set theory was established, the topology is thought as a structure o a set ad the theory of topology gives a basis to discuss covergece as well as the cotiuous deformatios of various objects The curret theory of topology cosists of geeral topology which is the foudatio to talk about covergece i various spaces ad cotiuity of maps betwee spaces, homotopy theory ad homology theory which defie various ivariats to distiguish spaces ad cofiguratios, ad applicatios of these geeral results to the iterestig objects such as maifolds 2 Covergece of Sequeces, Cotiuity of Maps, Geeral Topology 2 Metric Spaces ad the Covergece of Sequeces First we look at the spaces where there is a otio of distace I the -dimesioal Euclidea space, the distace betwee two poits = (,, ) ad y = ( y,, y ) is give by d(, y) = y = ( ) k k y = k distace fuctio d : satisfies the followig properties ) For, y, d(, y) = d( y, ) ; 2 This Ecyclopedia of Life Support Systems (EOLSS)

3 2) For, y, d(, y) ad d(, y) = if ad oly if = y ; 3) For, y, z, d(, z) d(, y) + d( y, z) By defiitio, a metric space is a set with a real-valued fuctio d : satisfyig the precedet properties () 3) There are may iterestig eamples of metric spaces, eg, the space B( S ) of real-valued bouded fuctios o a set S is a metric space by defiig d( f, g) = sup d( f( ), g( )) for f, g B( S) S For a metric space, we ca defie the ε eighborhood of a poit ad what is the coverget sequece For a positive real umber ε, the ε (ope) eighborhood Bε ( ) of is defied by Bε ( ) = { y d( y, ) < ε} A sequece (of poits) i a metric space is a mappig from to It is usually writte as { i A sequece { i i a metric space ( i ) is said to coverge to if for ay ε >, { i i Bε ( )} is a fiite set I this case, is called the limit of the sequece ad is writte as lim i i = If a sequece { i i a metric space coverges, it coverges to a uique poit i the metric space A subset A of a metric space is said to be closed if ay sequece { i of poits i A coverges to a poit, the A A sequece { i i a metric space is called a Cauchy sequece if for ay ε >, there is a umber k such that d( i, j) < ε for i, j k A coverget sequece is a Cauchy sequece But a Cauchy sequece may ot coverge i geeral For, oe may thik of the sequece {/ } o the ope half lie { > } If all Cauchy sequeces coverge, the metric space is called complete The -dimesioal Euclidea space is complete ad a closed subset A of a complete metric space is complete with respect to the distace fuctio restricted to A A map f : Y betwee metric spaces is cotiuous if for ay ad a positive real umber ε, there eists a positive real umber δ such that f ( Bδ( )) Bε( f( )) This is equivalet to say that if a sequece { i coverges to, the the sequece { f( i)} i coverges to f ( ) A ope set U of a metric space is defied to be a set U such that if U there is a positive real umber ε satisfyig Bε ( ) U That U is ope is equivalet to that the complemet U is closed By usig the otio of ope sets, a map f : Y betwee metric spaces is cotiuous if for ay ope set U of Y, f ( U) = { f( ) U} is a ope set of 22 Abstract Topology o Sets We ca thik of the cotiuity of the maps betwee spaces which are ot metric spaces First we otice that the set O of ope sets defied i a metric space has the Ecyclopedia of Life Support Systems (EOLSS)

4 followig properties: ) O ad O ; 2) k For a fiite subset { O i } i =, k, O, O i= i O ; 3) For ay subset { O λ } λ Λ O, O λ O λ Λ Now for ay set, we defie a topology of to be the set O of subsets of satisfyig the above three properties () 3) Elemets of O are called ope sets The set with a topology is called a topological space or simply a space Usig the otio of topology, we ca defie the otio of cotiuity of a map betwee topological spaces Give the set O of ope sets of ad the set O Y of ope sets of Y, a map f : Y is cotiuous if U O Y implies f ( U) O Whe the set O of ope sets of is give, the set N of ope eighborhoods of is defied as N = { O O O} Give the sets of ope eighborhoods N of ad N Y y of y Y, f : Y is cotiuous at if for ay ope Y eighborhood V N f ( ) there is a ope eighborhood U N such that f ( U) V This abstractio is useful to uderstad various otios cocerig covergece ad the cotiuity A closed set is the complemet of a ope set Let C be the set of closed sets of For a subset S of, the closure S of S is the smallest closed set cotaiig S, that is, S = C C For a metric space, the closure of a subset S cosists of all the C, S C poits which are the limits of sequeces i S which are coverget i For ay subset A of a topological space, A has the iduced topology defied by the set of ope sets give by { O A O O } With this topology, A is called a (topological) subspace of The iclusio map of ay subspace A to is a cotiuous map For two topological spaces ad Y, the direct product Y which is the set of the ordered pairs (, y) where ad y Y is a topological space The ope sets are the subsets of Y which is writte as a uio Uλ V λ Λ λ of direct products of ope sets U λ O ad V λ O Y ( λ Λ ) A map h: Y betwee topological spaces is called a homeomorphism if h is a bijectio ad h: Y ad its iverse map h : Y are cotiuous If there is a homeomorphism h: Y, ad Y are said to be homeomorphic For eample, the ope ball { d(, ) < } i the -dimesioal Euclidea space is homeomorphic to the -dimesioal Euclidea space Ecyclopedia of Life Support Systems (EOLSS)

5 The pricipal problem i topology theory is to determie whether two topological spaces are homeomorphic or ot To show that they are homeomorphic it is ecessary to show the way to costruct a homeomorphism betwee them To show that they are ot homeomorphic it is ecessary to defie a ivariat which has the same value if they are homeomorphic ad to show that the values of the ivariat are differet for the two spaces Homotopy groups ad homology groups eplaied later are topological ivariats ad used to show that two topological spaces are differet 23 Separatio Aioms ad Coutability Aioms The topological spaces usually treated have several ice properties which the metric spaces satisfy Such properties are also formulated i a abstract way A topological space is called a Hausdorff space, if for ay two poits ad y of, there are ope sets U ad V such that U, y V ad U V = Metric spaces are Hausdorff spaces I a Hausdorff space oe-poit set { } is a closed set as is the case i metric spaces A Hausdorff space is called a ormal space, if for ay two disjoit closed subsets C ad C of, there are ope sets U ad U such that C U, C U ad U U = I a ormal space, for ay two disjoit closed subsets C ad C of, there is a cotiuous fuctio f : [, ] such that C f = () ad C = f () This statemet is called Urysoh s Lemma Thus for a ormal space, there are a lot of real-valued cotiuous fuctios o ad oe ca distiguish closed sets by them A subset B of the set N of ope eighborhoods of is called a base of N if for ay elemet U of N there is V B such that V U If there is a coutable base B of N at every poit, the topological space is said to satisfy the first aiom of coutability If is a metric space, the topology of satisfies the first aiom of coutability A subset B of O is a base of O if ay elemet U of O is writte as U = V V B, V U If there is a coutable base B of O, is said to satisfy the secod aiom of coutability If satisfies the secod aiom of coutability, the it of course satisfies the first aiom of coutability For the direct product Y of topological spaces ad Y, { U V U O, V O Y} is the base of O Y If ad Y satisfy the secod aiom of coutability, the Y satisfies it The -dimesioal Euclidea space is a metric space, hece it satisfies the first aiom of coutability also satisfies the secod aiom of coutability It is easy to see that ay subspace of with the iduced metric satisfies both aioms of coutability Ecyclopedia of Life Support Systems (EOLSS)

6 - - - TO ACCESS ALL THE 2 PAGES OF THIS CHAPTER, Visit: Bibliography Ryszard Egelkig, (989) Geeral topology, Secod editio, Sigma Series i Pure Mathematics, 6 Helderma Verlag, Berli, viii+529 pp ISBN: [This is a reasoably complete epositio of geeral topology] Ryszard Egelkig ad Karol Sieklucki, (992) Topology: a geometric approach, Sigma Series i Pure Mathematics, 4 Helderma Verlag, Berli, +429 pp ISBN: [This is a itroductio to geeral topology as well as to geometric topology] James R Mukres, (984) Elemets of algebraic topology Addiso-Wesley Publishig Compay, Melo Park, CA, i+454 pp ISBN: [This is a tet book o the homology ad cohomology theory] William Fulto, (995) Algebraic Topology: A First Course (Graduate Tets i Mathematics, Vol 53) Spriger, ISBN [This book begis with elemetary itroductio of homotopy theory ad homology ad cohomology theory ad treats several importat applicatios] William S Massey, (99) A basic course i algebraic topology (Graduate Tets i Mathematics, Vol 27) Spriger-Verlag, New York, vi+428 pp ISBN: [This is a tet book o the homology ad cohomology theory] Joh Milor, (965) Lectures o the h-cobordism theorem (with L Siebema ad J Sodow), Priceto U Press [This book eplais how oe ca use the Morse theory to ivestigate the topology of maifolds] S Smale, (96) Geeralized Poicaré s cojecture i dimesios greater tha four, Aals of Mathematics vol [This paper gives the proof of the Poicaré cojecture for the spheres of dimesio greater tha 4] M H Freedma, (982) The topology of four-dimesioal maifolds, Joural of Differetial Geometry vol 7, [This paper gives the proof of the Poicaré cojecture for the sphere of dimesio 4] Joh W Morga, (25) Recet progress o the Poicaré cojecture ad the classificatio of 3- maifolds, Bulleti (New Series) of the America Mathematical Society Volume 42, Number, Pages [This is a epository paper o the proof by Perelma of the Poicaré cojecture for the sphere of dimesio 3] W B Raymod Lickorish, (997) Itroductio to kot theory, (Graduate Tets i Mathematics, Vol 75) Spriger-Verlag, New York, +2 pp ISBN [This book eplais a umber of ivariats for kots ad liks kow before the begiig of 99 s] William Measco ad Morwe Thistlethwaite (Editors), (25) Hadbook of kot theory, Elsevier BV, Amsterdam, +492 pp ISBN [This book cotais the progress i kot theory made at the begiig of 2st cetury] Biographical Sketch Takashi Tsuboi, bor 953 i Hiroshima, Japa Educatio: Ecyclopedia of Life Support Systems (EOLSS)

7 BS i Mathematics, the Uiversity of Tokyo, Japa (March, 976) MS i Mathematics, the Uiversity of Tokyo, Japa (March, 978) PhD i Mathematics, the Uiversity of Tokyo, Japa (February, 983) Positios held: Assistat, Departmet of Mathematics, the Uiversity of Tokyo, Japa (April, 978--March, 985) Assistat Professor, Departmet of Mathematics, the Uiversity of Tokyo, Japa (April, September, 993) Professor, Graduate School of Mathematical Scieces, the Uiversity of Tokyo, Japa (October, 993 to date) Ecyclopedia of Life Support Systems (EOLSS)