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Part III Lecture Notes




    3-manifolds is a subject thats had extraordinary progress in the last decade, as withthe geometrization conjecture proof. I wont be able to prove these theorems, but I wantto get us to the point where we understand what they mean. The topology of manifoldsis kind of like the first line of Anna Karenina. The high-dimensional manifolds are happyand understood, but 3- and 4folds have very much their own unique flavors.

    1. Basics

    1.1. Foundations.

    1.1.1. Surfaces. Here are 3 possible definitions of what it means for a compact metrizablespace S to be a surface.

    (Topological) Every p S has an open neighborhood Up such that Up is homeo-morphic to the open unit ball B2 R2. (Smooth) Same as topological, but the transition functions p 1q : B2 B2

    are smooth. (Piecewise Linear/Combinatorial) A quotient of a disjoint union of triangles by

    identifications of edges in pairs.

    Its a nontrivial fact that these three definitions are equivalent in the following sense:if S is a topological surface, there exists a smooth surface S and a homeomorphismbetween them; if smooth surfaces are homeomorphic, then theyre diffeomorphic; if Sis topological theres a combinatorial S homeomorphic to S, and if two combinatorialsurfaces are homeomorphic, then theyre isomorphic by a piecewise linear map aftersubdivision.

    A fact we know better is the classification of surfaces as connected sums of the spherewith tori and real projective planes. A note about defining connected sums: use transi-tion maps on the inverse image of the circle of radius 1/2 around a point in each surface,and to have the connected sum be well defined remember to suppose the surfaces areconnected.

    1.1.2. 3-manifolds. The first direction of approach is this: which of the facts aboutsurfaces is going to go through for 3-manifolds? The first two definitions are identicalup to dimension. For the third: suppose Y is obtained by starting with a finite set oftetrahedra and identifying faces in pairs. This wont necessarily give a 3-manifold.

    Definition 1.1. Given v Y as above, definelk(v) = {x Y : x T, v is a vertex of T, dT (x, v) = 1/10}

    Date: 10 Oct 2013.



    So, by construction, lk(v) is a combinatorial surface, and v has a neighborhood homeo-morphic to the cone on lk(v).

    Proposition 1.2. Y is a topological 3-manifold iff the link of v is S2 for every vertexv.

    The proof will be on the example sheet. That aside, we can now make a

    Definition 1.3. Y is a combinatorial 3-manifold if its given by identification of tetrahedra-faces pairwise such that every link is a sphere.

    Now that weve fixed our definitions, we can state

    Theorem 1.4 (Moise, 1952). All three definitions are the same, in the same sense asfor surfaces.

    This is very much a non-trivial theorem. We remark that it does not hold for n-manifolds even when n = 4. We can define connected sum in the same way, but theobvious analogue of the classification theorem for surfaces is totally false: the semigroupof 3-manifolds under connected sum is not finitely generated.

    1.1.3. Manifolds with boundary. Let Y is obtained by identifying some, but not neces-sarily all, faces of a finite set of tetrahedra in pairs.

    Definition 1.5. Y , the union of the non-identified spaces, is a combinatorial surface.If the appropriate condition holds about links, specifically each link is either S2 or D2,Y is called a 3-manifold with boundary.

    There arent any 3-manifolds without boundary that embed into R3, but there areplenty with boundary. For instance, we can solidify any embeddable surface to get thingslike the double solid torus. This always gives half a 3-manifold, as follows:

    Definition 1.6. The double D(Y ) = Y Y Y of a 3-manifold with boundary Y isalways a closed 3-manifold. e.g. S3 = D(D3), D(S1 D2) = S1 S2. Question: whatsthe double of the solid double torus?

    1.1.4. Submanifolds. Suppose Y is a combinatorial (resp smooth) 3-manifold andM Yis a 1- or 2-manifold in the subspace topology. We say M is nicely embedded if its asubcomplex (smooth submanifold) of Y . If Y is a topological 3-manifold and we haveM Y a < 3-manifold, we say M is tamely embedded if theres a homeomorphismY ' Y , Y combinatorial, sending M to a nicely embedded submanifold.

    So whats the point of this definition? Theres a

    Theorem 1.7. If M Y is a tame submanifold, it has an open regular neighborhood(M) Y such that (M) ' the open disk bundle in the normal bundle of M . (dontyou have to make Y smooth to talk about the normal bundle?)

    Here are some non-tame examples.

    Example 1.8. A geometric series of non-closed knots that wraps back around to itself isa wild 1-submanifold of 3. Alexanders horned sphere is a wild 2-submanifold.


    2. Examples

    2.1. Heegard splittings.

    Definition 2.1. Suppose Y1, Y2 are compact 3-manifolds and Yi is nonempty. Takepi Yi and Ui upper-hemisphereB3 neighborhoods of pi. Then the boundary connectedsum Y1#Y2 is Y1 \ U1 D2 Y2 \ U2.

    For this to work we need connected boundaries and something else...checking links?

    Definition 2.2. Handlebody H0 = D3, H1 = S

    1 D2, Hg = #gS1 D2. If is ahomeomorphism from g = Hg to itself, we can define Y = Hg Hg the Heegaardsplitting for Y.

    Example 2.3. Doubling. If i are self-homeomorphisms of Si which restrict to identityon open Euclidean neighborhoods of pi then 1#2 is well-defined.

    Lemma 2.4. Y1#2 = Y1#Y2.

    Proof. Let Hi be a handlebody bounded by Si and Hi = Hi \ Vi for Vi half-ball neigh-

    borhoods of pi in Hi. Then H1#H2 =: H is a handlebody bounded by S1#S2.Now

    Y1#2 = H 1#2 H = (H 1 D2 H 2) 1#2 (H 1 D2 H 2)= (H 1 varphi1 H 1) D2D2 (H 2 2 H 2) = (Y1 \B3) S2 (Y2 \B3) = Y1#Y2

    2.2. Lens spaces. Given A GL2(Z), call A the induced self-homeomorphism of T 2.On the solid torus take two circles through P , ` = S1P and m = D2 at P . If p, q Zare relatively prime then there exist a, b with ap bq = 1 so A =

    (b pa q

    ) SL2(Z).

    Now we have

    Definition 2.5. The lens space L(p, q) = YA . We claim L(p, q) does not depend onthe choice of a and b.

    Lemma 2.6. pi1(L(p, q)) = Z/p.

    Proof. Use Seifert-van Kampen. pi1(Hi) = `i ' Z. Let ci : T 2 Hi identify T 2 withthe boundary of the second S1D2. Then c1(m) = `p1, c1(`) = `b1, c2(m) = 0, c2(`) = `2,giving pi1(H) = `1, `2, `p1 = 1, `b2 = `1. 2.3. Mapping Class Groups.

    Definition 2.7. Two homeomorphisms 0, 1 : X X are isotopic if theres a map : X I X going from 0 to 1 such that t is a homeomorphism for all t.

    We remark that isotopy is an equivalence relation.

    Proposition 2.8. If 0 is isotopic to 1, then Y0 ' Y1.Proof. Observe that Hg ' HgfgI, f : g0id g. So Yi = [HgfgI]iHg.If is an isotopy from 0 to 1 then F : Y0 Y1 , given by F |Hg = id, F |gI sending(x, t) 7 (1(0(x)), t) is a homeomorphism.


    Let NX Homeo(X) be the set of homeomorphisms isotopic to idX . Then NX is anormal subgroup, since if is an isotopy from 0 to idX then

    1 is an isotopy from0

    1 to idX 1 = idX for any Homeo(X). Then we giveDefinition 2.9. The mapping class group MCG(X) = Homeo(X)/NX .

    So Y only depends on the image of in the mapping class group.

    Example 2.10. GL2(Z) Homeo(T 2) intersects NT 2 trivially since any NT 2 actstrivially on pi1(T

    2). So GL2(Z) MCG(T 2). In fact this is an equality.On the other hand, on Thursday well discuss the mapping class group of a higher-

    genus surface is still a subject for research- its much more interesting and much biggerthan GL2(Z). One justification for the existence of many distinct 3-manifolds is exactlythe size of these mapping class groups.

    Recall we just saw that linear maps define an embedding GL2(Z)MCG(T 2).Identify S1 ' [0, 1]/ and map S1 I to itself via T (x, t) = (x+ t, t). Draw a picture

    on the fundamental square of the cylinder. Note that T fixes S1 I. It is isotopic tothe identity of S1 I, but not through homeomorphisms that fix S1 I. Now supposeS is an oriented surface with a simply closed curve and () a regular neighborhood.Choose an orientation-preserving homeomorphism : () S1 I. Note we neededS to be orientable since otherwise () could be a Mobius strip.

    Definition 2.11 (Dehn Twists). Define the positive Dehn twist around by

    : S S, = 1TAway from () extend by the identity.

    Proposition 2.12. The class represented by in the mapping class group of S doesnot depend on the choice of or that of ().

    Example 2.13. Consider S = T 2 = m `, where the simple closed curves m, ` are theedges of the fundamental polygon. We see m fixes m. But we claim that it sends `to something like {0} [0, 1/3] [2/3, 1] {(x, 1/3x + 1/3), x I} when we identify() with S1 [1/3, 2/3]. If we similarly move (`) into a vertical strip in the centerit should send m into what looks like a hyperbola on the square. Some more detail:[m(`)] = [m] + [`] and [`(m)] = [m] [`].

    So in summary m acts on H1(T2) = Z2 by

    (1 10 1

    )=: A and ` by

    (1 01 1

    )=: B.

    Observe that A and B generate SL2(Z). In fact, if MCG+(S) is the subgroup generated

    by orientation-preserving maps, then MCG+(S) is generated by Dehn twists on curvein S. Can we get all of it if we permit orientation-reversing Dehn twist?

    2.4. Back to 3-manifolds.

    Example 2.14 (Manifolds that fiber over S1). Let : S S be a homeomorphismof a connected surface. Define Z = S I/ where (x, 1) ((x), 0). The sameargument as for Heegaard splitting shows that Z only depends on the equivalence classof MCG(S). Observe that we have a map pi : Z S1 mapping (x, t) 7 t I/ .Observe that pi1(t) is S. We say that Z fibres over S1 with fibre S