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Surface Topology Surface Topology Tara E. Brendle July 22, 2011 Contents 1Lecture1:Whatisasurface? 3 1.1 Definitionofasurface ................................ .3 1.2 Surfaceswithboundary .............................. .. 4 1.3 Closedsets ....................................... 4 1.4 Compactness ...................................... 4 1.5 Connectedness,path-connectedness . ......... 6 1.6 Wherewe’reheading.................................. 7 2Lecture2:Quotients 8 2.1 Quotientsassetswithoutmetrics . .... 8 2.2 Whathappenstothemetric? .. .. .. .. .. .. .. .. .. .. .. .. .. 8 2.3 Pathification ...................................... 8 2.4 Examples ........................................ 9 3Grouptheory 11 3.1 Groupsandhomomorphisms .. .. .. .. .. .. .. .. .. .. .. .. .11 3.2 Groupactions...................................... 12 3.3 Forming quotients of spaces via group actions . ...... 13 4Lecture4:Eulercharacteristic 14 4.1 Euler characteristic: definition/examples . ...... 14 4.2 Eulercharacteristicandgenus. .... 15 5Lecture5:FundamentalGroups 18 5.1 Homotopy........................................ 18 5.2 FundamentalGroups................................ .. 18 5.3 Basepoints and functoriality . ... 19 6Lecture6:Functoriality,CoveringSpacesandDeckTransformations 21 6.1 Functoriality ...................................... 21 6.2 First examples: Convex sets in Rn; the circle S1 .................. 21 6.3 Coveringspaces.................................... .22 6.4 Decktransformations............................... ... 22 6.5 Liftingpathsandhomotopies . .. 23 6.6 Finishing the proof of the Isomorphism Theorem . ..... 24 1 6.7 TheTorus,Again.................................... 25 7Lecture7:MoreonFundamentalGroupsandCoveringSpaces 26 7.1 HomotopyEquivalence................................ .26 7.2 Actionofthedeckgrouponthecover . .... 27 7.3 Fundamental groups of surfaces given as plane models . ........ 28 8Lecture8:JordanCurveTheorem 29 8.1 GraphTheoryCheatSheet ............................ .. 29 8.2 PolygonalJordanCurveTheorem. .... 30 8.3 GeneralJordanCurveTheorem. .... 31 9Lecture9:TriangulabilityofSurfaces 33 9.1 FinishingupJordanCurveTheorem . .. 33 9.2 Corollaries of Jordan-Schoenflies . .... 34 9.3 Proof of Triangulability, assuming Jordan-Schoenflies. ........ 35 9.4 Well-definednessofconnectedsum . .... 36 9.5 Finishing Classification . 36 10 Lecture 10: Proof of Triangulability of Surfaces 37 10.1 Well-definednessofconnectedsum . ..... 37 10.2 Finishing Classification . .37 10.3 Classification of simple closed curves on surfaces . ....... 38 11 Lecture 11: Basics of curves on surfaces 40 11.1EssentialCurves .................................. .. 40 11.2 Algebraic intersection . .. 40 11.3 Geodesicrepresentatives . ..... 40 11.4Torusexample ..................................... 41 11.5Bigoncriterion ..................................... 41 12 Lecture 12: Mapping Class Groups I 43 12.1 PuncturesversusBoundaryComponents . ........ 43 12.2Finiteorderexamples ................................ .43 12.3Somewarmupexamples ............................... .43 13 Lecture 13: Mapping Class Groups II 45 13.1Dehntwists....................................... 45 13.2Torus,again...................................... .46 13.3Exactsequences.................................. ... 46 14 Lecture 14: Mapping Class Groups III 47 14.1Generatingsets................................... .. 47 14.2Thecurvecomplex .................................. .47 14.3 Variationsonthecurvecomplex . ... 47 14.4 Generation of Mod(S).................................48 2 Acknowledgement: This lecture follows two excellent treatments of the basics of surface topology: Armstrong’s Basic Topology and Goodman’s Beginning Topology. 1Lecture1:Whatisasurface? Note that all “spaces” are metric spaces. Those with a background in topology can think of these as topological spaces, with the topology generated by the open sets induced by the metric. 1.1 Definition of a surface The OED’s first definition of a surface is: “The outermost boundary (or one of the boundaries) of any material body, immediately adjacent to the air or empty space, or to another body.” Compare with the mathematical definition: A surface S is a metric space which is locally homeomorphic to a disk. In other words, for every point p ∈ S, there exists an open set U ⊆ S such that p ∈ U and such that U is homeomorphic to the open disk {(x, y) | x2 + y2 < 1}. A topological space X is called Hausdorff, if for every a, b ∈ X with a #= b, there exist disjoint open sets U(a),U(b) containing a, b, respectively. In a topology course, a surface is usually defined with the word “Hausdorff” replacing “metric”, which gives a slightly more general notion of a surface. Note that every metric space is automatically Hausdorff, since if ! d(a, b)=!,thenwecantakeU(a),U(b) each to be balls of radius less than 2 centered at a, b, respectively. Note to those with topology background: a compact Hausdorffspace admits a metric if and only if it’s second countable. Also, a locally metrizable space is metrizable iffit is Hausdorff and paracompact. Examples. • The plane R2 with the standard metric. • The sphere S2: : by symmetry it’s enough to find a neighborhood of (0, 0, 1) homeomor- phic to a disk. Then take, say, all points with z>0 and show that projection to xy-plane is a homeomorphism. • The torus T 2, version one: surface of revolution in R2 • The torus T 2, version two: S1 × S1 in R4 Nonexample. • To see why being locally homeomorphic to a disk is not a sufficient condition: Start with the plane. Remove (0, 0) and replace it with two “copies”. 3 1.2 Surfaces with boundary A surface with boundary S is a metric space such that for every point p ∈ S, there exists an open set U ⊆ S such that p ∈ U and such that U is homeomorphic to an open disk or to the half disk {(x, y) | x2 + y2 < 1andx ≥ 0}.Ifp has a neighborhood of the first type it is an interior point, otherwise p is a boundary point. Examples of surfaces with boundary. • The disk D2 = {(x, y) | x2 + y2 ≤ 1} • The annulus A = {(x, y) | 1 ≤ x2 + y2 ≤ 2} Exercise 1.1. Which of the following are surfaces? Surfaces with boundary? 1. {(x, y, z) ∈ R3 | x2 + y2 + z2 =1and z ≥ 0} 2. {(x, y, z) ∈ R3 | 4x2 + y2 +9z2 =1} 3. {(x, y, z) ∈ R3 | x =0or z =0} 1.3 Closed sets A subset of a space is closed if its complement is open. Exercise 1.2. Think of an example of a subset of R2 with the usual metric which is both open and closed, and one which is neither open nor closed. Let S be a subset of a space X.Thenp ∈ X is a limit point of S (sometimes known as a point of accumulation) if every open set which contains p also contains at least one point of S −{p}. Exercise 1.3. Determine the limit points of [0, 1) in R. 1 Exercise 1.4. Determine the limit points of the set of points of the form n in R,forn ∈ N. The union of a set S and its limit points is called the closure of S. Exercise 1.5. Asetisclosedifandonlyifitcontainsallitslimitpoints. Therefore a set is closed if and only if it is equal to its closure. 1.4 Compactness A space is compact if every open cover has a finite subcover. That is, if X = ∪α∈AVα,where n Vα is open for all α in some (possibly infinite) index set A,thenX = ∪i=1Vαi for some finite subset of the Vα’s. Note about terminology. Just to ensure constant confusion, mathematicians often use the term “surface” to refer to both surfaces and surfaces with boundary as defined above, and use the term “closed surface” to mean “compact surface without boundary”, i.e., as opposed to a set that’s closed in the sense that its complement is open. 4 Theorem 1.6 (Heine-Borel for R). Aclosedintervalofthereallineiscompact. Proof. Let C be an open cover of [a, b]. Consider the set X of all x ∈ [a, b]suchthat[a, x] is contained in a finite subcover of C. First note that X has a least upper bound s since it is nonempty and bounded above. We will show that s ∈ X and that s = b.LetU ∈ C be an open set containing s.Ifs<bwe can choose ! such that (s − !, s + !) ⊆ U.(Ifs = b, we can still assume (s − !, s] ⊆ U.) Now, note that if x ∈ X and if a ≤ y ≤ x,theny ∈ X. Then, by definition of s,wemust ! C" C ! have that s − 2 ∈ X, i.e., there is a finite subcover of for the interval [a, s − 2 ]. But then C" ∪ U is a finite cover of [a, s], and hence s ∈ X. C" ! Finally, to see that s = b, we observe that if s<b,then ∪ U actually covers [a, s + 2 ], which contradicts the fact that s is a least upper bound for X. Note that this proof relied heavily on the completeness property of the real numbers, specif- ically that a nonempty set of real numbers which has an upper bound has a least upper bound. The next two results aren’t hard to prove, but they are important enough to call them theorems. Theorem 1.7. AclosedsubsetC of a compact space X is compact. Proof. Let F be an open cover of C.ThenF ∪ (X − C) is an open cover of X,andhencehasa finite subcover F".ThenF" (or possibly F" − (X − C)) is the finite subcover of C we are looking for. Theorem 1.8. Prove that if f : X → Y is continuous, and then f(A) is compact if A is compact. Proof. Without loss of generality, we may assume that X is compact and that f is surjective. Let F be an open cover of Y . Then since f is continuous, {f −1(U) | U ∈ F} is an open −1 −1 cover of X and has a finite subcover of the form f (U1),...,f (Un). Since f is surjective, −1 f(f (U)) = U for all U,hence{U1,...,Un} is the finite subcover of Y we require. Theorem 1.9 (Heine-Borel for Rn). AsubsetofRn is compact if and only if it is closed and bounded. We’ll leave the “forward” direction as a series of exercises. To prove the converse, we’ll need the following proposition: Proposition 1.10. AfiniteproductofclosedintervalsinRn is compact. Proof of Proposition. We’ll show that S =[a, b] × [c, d] is compact and leave the general case as an exercise (it’s virtually identical). Let F be an arbitrary open cover of S. Without loss of generality, we may assume F is consists of sets of the form U × V ,whereU and V are both open intervals. (Exercise: explain why this is true.) Now, let (x, y) be a point in S. Since F also covers {x}×[c, d], and since {x}×[c, d] is homeomorphic to [c, d], an interval, there must be a finite subcover of {x}×[c, d] consisting of x x x elements of the form Ui × Vi ,fori =1,...,m(x).
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