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1
SMSTC Geometry and Topology 2011{2012
Lecture 7
The classication of surfaces
Andrew Ranicki (Edinburgh)
Drawings: Carmen Rovi (Edinburgh)
24th November, 2011
2
Manifolds
IAnn-dimensional manifoldMis a topological space such that eachx∈Mhas an open neighbourhoodU⊂M homeomorphic ton-dimensional Euclidean spaceRn U ∼=Rn. I Strictly speaking, need to include the condition thatMbe Hausdor and paracompact = every open cover has a locally nite renement. I
Calledn-manifoldfor short.
I Manifolds are the topological spaces of greatest interest, e.g. M=Rn. I
Study of manifolds initiated by Riemann (1854).
I
Asurfaceis a 2-dimensional manifold.
I Will be mainly concerned with manifolds which are compact = every open cover has a nite renement. 3
Why are manifolds interesting?
ITopology.
I
Dierential equations.
I
Dierential geometry.
I
Hyperbolic geometry.
I
Algebraic geometry. Uniformization theorem.
I
Complex analysis. Riemann surfaces.
I
Dynamical systems,
I
Mathematical physics.
I
Combinatorics.
I
Topological quantum eld theory.
I
Computational topology.
I
Pattern recognition: body and brain scans.
I 4 Examples ofn-manifoldsIThen-dimensional Euclidean spaceRnIThen-sphereSn. I
Then-dimensional projective space
RP n=Sn/{z∼ -z}. I Rank theorem in linear algebra. IfJ:Rn+k→Rkis a linear map of rankk(i.e. onto) thenJ-1(0) = ker(J)⊆Rn+k is ann-dimensional vector subspace. I Implicit function theorem. The solutions of dierential equations are generically manifolds. Iff:Rn+k→Rkis a dierentiable function such that for everyx∈f-1(0) the Jacobiank×(n+k) matrixJ= (∂fi/∂xj) has rankk, then
M=f-1(0)⊆Rn+k
is ann-manifold. I In fact, everyn-manifoldMadmits an embeddingM⊆Rn+k for some largek. 5
Manifolds with boundary
IAnn-dimensional manifold with boundary(M,∂M⊂M) is a pair of topological spaces such that (1)
M\∂Mis ann-manifold called theinterior,
(2) ∂Mis an (n-1)-manifold called theboundary, (3) Eachx∈∂Mhas an open neighbourhoodU⊂Msuch that I
A manifoldMisclosedif∂M=∅.
I The boundary∂Mof a manifold with boundary (M,∂M) is closed,∂∂M=∅. I
Example(Dn,Sn-1) is ann-manifold with boundary.
I
ExampleThe product of anm-manifold with boundary
(M,∂M) and ann-manifold with boundary (N,∂N) is an (m+n)-manifold with boundary 6 The classication ofn-manifolds I.IWill only consider compact manifolds from now on.
IA function
i: a class of manifolds→a set ;M7→i(M) is atopological invariantifi(M) =i(M′) for homeomorphic M,M′. Want the set to be nite, or at least countable. I
Example 1The dimensionn>0 of ann-manifoldMis a
topological invariant (Brouwer, 1910). I Example 2The number of componentsπ0(M) of a manifold
Mis a topological invariant.
I
Example 3The orientabilityw(M)∈ {-1,+1}of a
connected manifoldMis a topological invariant. I Example 4The Euler characteristicχ(M)∈Zof a manifold
Mis a topological invariant.
I Aclassicationofn-manifolds is a topological invarianti such thati(M) =i(M′) if and only ifM,M′are homeomorphic. 7 The classication ofn-manifolds II.n= 0,1,2,...IClassication of 0-manifoldsA 0-manifoldMis a nite set of points. Classied byπ0(M) = no. of points>1. I Classication of 1-manifoldsA 1-manifoldMis a nite set of circlesS1. Classied byπ0(M) = no. of circles>1. I Classication of 2-manifoldsClassied byπ0(M), and for connectedMby the fundamental groupπ1(M). Details to follow! I Forn62 homeomorphism⇐⇒homotopy equivalence. I It is theoretically possible to classify 3-manifolds, especially after the Perelman solution of the Poincare conjecture. I It is not possible to classifyn-manifolds forn>4. Every nitely presented group is realized asπ1(M) =⟨S|R⟩for a
4-manifoldM. The word problem is undecidable, so cannot
classifyπ1(M), let aloneM. 8 9
How does one classify surfaces?
I(1) Every surfaceMcan betriangulated, i.e. is
homeomorphic to a nite 2-dimensional cell complex M c 0D
0∪∪
c 1D
1∪∪
c 2D 2. I (2) Every connected triangulatedMis homeomorphic to a normal form
M(g) orientable, genusg>0,
N(g) nonorientable, genusg>1
I (3) No two normal forms are homeomorphic. I Similarly for surfaces with boundary, with normal forms
M(g,h),N(g,h) with genusg, andhboundary circles.
I History: (2)+(3) already in 1860-1920 (Mobius, Cliord, van Dyck, Dehn and Heegaard, Brahana). (1) only in the 1920's (Rado, Kerekjarto). Today will only do (3), by computingπ1 of normal forms. 10 A page from Dehn and Heegaard'sAnalysis Situs(1907) 11
The connected sum I.
IGiven ann-manifold with boundary (M,∂M) withM connected use any embeddingDn⊂M\∂Mto dene the puncturedn-manifold with boundary (M0,∂M0) = (cl.(M\Dn),∂M∪Sn-1). I Theconnected sumof connectedn-manifolds with boundary (M,∂M), (M′,∂M′) is the connectedn-manifold with boundary (M#M′,∂(M#M′)) = (M0∪Sn1M′0,∂M∪∂M′). Independent of choices ofDn⊂M\∂M,Dn⊂M′\∂M′. I
IfMandM′are closed then so isM#M′.
12
The connected sum II.
IM M'
M # M'
I The connected sum # has a neutral element, is commutative and associative: (i)M#Sn∼=M′, (ii)M#M′∼=M′#M, (iii) (M#M′)#M′′∼=M#(M′#M′′). 13
The fundamental group of a connected sum
IIf (M,∂M) is ann-manifold with boundary andMis connected thenM0is also connected. Can apply the
Seifert-van Kampen Theorem to
M=M0∪Sn1Dn
to obtain
1(M) =π1(M0)∗1(Sn1){1}=
1(M0) forn>3
1(M0)/⟨∂⟩forn= 2
with⟨∂⟩▹ π1(M0) the normal subgroup generated by the boundary circle∂:S1⊂M0. I Another application of the Seifert-van Kampen Theorem gives
1(M#M′) =π1(M0)∗1(Sn1)π1(M′0)
1(M)∗π1(M′) forn>3
1(M0)∗Zπ1(M′0) forn= 2.
14
Orientability for surfaces
ILetMbe a connected surface, and letα:S1→Mbe an injective loop. I αisorientableif the complement is not connected, in which case it has 2 components. I αisnonorientableif the complementM\α(S1) is connected. I DenitionMisorientableif everyα:S1→Mis orientable. I
Jordan Curve TheoremR2is orientable.
I
ExampleThe 2-sphereS2and the torusS1×S1are
orientable. I DenitionMisnonorientableif there exists a nonorientable α:S1→M, or equivalently if Mobius band⊂M. I ExampleThe Mobius band, the projective planeRP2and the
Klein bottleKare nonorientable.
I RemarkCan similarly dene orientability for connected n-manifoldsM, usingα:Sn-1→M,π0(M\α(Sn-1)). 15 The orientable closed surfacesM(g)I.IDenitionLetg>0. Theorientable connected surface with genusgis the connected sum ofgcopies ofS1×S1
M(g) = #
g(S1×S1) I
ExampleM(0) =S2, the 2-sphere.
I
ExampleM(1) =S1×S1, the torus.
I
ExampleM(2) = the 2-holed torus, by Henry Moore.
16
The orientable closed surfacesM(g)II.M(1)
M(0) M(2) M(g) 17 The nonorientable surfacesN(g)I.ILetg>1. Thenonorientable connected surface with genusgis the connected sum ofgcopies ofRP2
N(g) = #
gRP2 I
ExampleN(1) =RP2, the projective plane.
I
Boy's immersion ofRP2inR3
(in Oberwolfach) 18
The nonorientable closed surfacesN(g)II.N(g)
Projective plane = N(1)
Klein bottle = N(2)
19
The Klein bottle
IExampleN(2) =Kis the Klein bottle.IThe Klein bottle company 20
The classication theorem for closed surfaces
ITheoremEvery connected closed surfaceMis homeomorphic to exactly one of
M(0),M(1), ... ,M(g) = #
gS1×S1, ...(orientable)
N(1),N(2), ... ,N(g) = #
gRP2, ...(nonorientable) I
Connected surfaces are classied by the genusgand
orientability. I Connected surfaces are classied by the fundamental group :
1(M(g)) =⟨a1,b1,a2,b2,...,ag,bg|[a1,b1]...[ag,bg]⟩
1(N(g)) =⟨c1,c2,...,cg|(c1)2(c2)2...(cg)2⟩
I Connected surfaces are classied by the Euler characteristic and orientability
χ(M(g)) = 2-2g, χ(N(g)) = 2-g.
21
The punctured torus I.
IThe computation ofπ1(M(g)) forg>0 will be by induction, using the connected sum
M(g+ 1) =M(g)#M(1)
I So need to understand the fundamental group of the torus
M(1) =T=S1×S1and the puncture torus (T0,S1).
I
Clear fromT=S1×S1thatπ1(T) =Z⊕Z.
I Can also get this by applying the Seifert-van Kampen theorem toM(1) =M(1)#M(0), i.e.T=T0∪S1D2. I
The punctured torus
(T0,∂T0) = (cl.(S1×S1\D2),S1) is such thatS1∨S1⊂T0is a homotopy equivalence. 22
The punctured torus II.
IThe inclusion∂T0=S1⊂T0induces
1(S1) =Z→π1(T0) =π1(S1∨S1) =Z∗Z=⟨a,b⟩;
17→[a,b] =aba-1b-1.
ITorus
b b a a Iquotesdbs_dbs17.pdfusesText_23
5 Topology