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Topology and Geometry of 2 and 3 Dimensional Manifolds

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Topology and Geometry of 2 and 3 Dimensional Manifolds Topology and Geometry of 2 and 3 dimensional manifolds Chris John May 3, 2016 Supervised by Dr. Tejas Kalelkar 1 Introduction In this project I started with studying the classification of Surface and then I started studying some preliminary topics in 3 dimensional manifolds. 2 Surfaces Definition 2.1. A simple closed curve c in a connected surface S is separating if S n c has two components. It is non-separating otherwise. If c is separating and a component of S n c is an annulus or a disk, then it is called inessential. A curve is essential otherwise. An observation that we can make here is that a curve is non-separating if and only if its complement is connected. For manifolds, connectedness implies path connectedness and hence c is non-separating if and only if there is some other simple closed curve intersecting c exactly once. 2.1 Decomposition of Surfaces Lemma 2.2. A closed surface S is prime if and only if S contains no essential separating simple closed curve. Proof. Consider a closed surface S. Let us assume that S = S1#S2 is a non trivial con- nected sum. The curve along which S1 n(disc) and S2 n(disc) where identified is an essential separating curve in S. Hence, if S contains no essential separating curve, then S is prime. Now assume that there exists a essential separating curve c in S. This means neither of 2 2 the components of S n c are disks i.e. S1 6= S and S2 6= S . Lemma 2.3. The torus, the projective plane and the sphere are all prime surfaces. Lemma 2.4. The only closed prime surfaces are the torus, the projective plane and the sphere. Theorem 2.5. (The classification of closed compact surfaces) Every closed compact con- nected surface is homeomorphic to a sphere or a connected sum of tori or a connected sum of projective planes. Example The figure below gives the prime decomposition of a 2-torus. Figure 1: Prime decomposition of a 2-torus Definition 2.6. The number of tori in the connected sum is called the genus of a orientable surface. For non orientable surfaces, the number of projective planes in the connected sum is called the genus. Corollary 2.7. For a surface S of genus g, χ(S) = 2 − 2g. For a non orientable surface S of genus g, χ(S) = 1 − g: Here χ(S) is the Euler characteristic of the surface S. There is one another way of decomposition of surfaces, the pants decomposition. Definition 2.8. A pants decomposition of a compact surface S is a collection of pairwise disjoint simple closed curves fc1; :::; cng such that each component of S n (c1 [ c2 [ ::: [ cn) is a pair of pants. Two pants decompositions are equivalent if they are isotopic. Pants decomposition of surfaces are not unique. The following figure contains different decomposition of 2-torus. Figure 2: Pants decomposition of a 2-torus Figure 3: Pants decomposition of a 2-torus 2 2.2 Covering spaces and branched covering spaces Definition 2.9. Let p : E ! B be a continuous map. The open set U ⊂ B is said to be −1 evenly covered if p (U) is a disjoint union of open sets fUαg such that 8α; p jUα : Uα ! U is a homeomorphism. Definition 2.10. Let E be a manifold, B a connected manifold, and p : E ! B a contin- uous map. The triple (E; B; p) is a covering if 8b 2 B there is an open set U ⊂ B with b 2 U that is evenly covered. The map p is called the covering map. Example The triple (R; S1; p), where p : R ! S1 is given by p(x) = e2πix is a covering. Example The triple (R2; T2; p), where p : R2 ! T2 is given by p(x) = (e2πix; e2πiy), is a covering. If suppose p : E ! B is a covering map, then p is a local homeomorphism, but p a local isomorphism may not be a covering map. For example, consider p : R+ ! S1 given by p(x) = e2πix. Definition 2.11. Let (E; B; p) be a covering space. A homeomorphism t : E ! E is a covering transformation if p ◦ t = p: Example For n 2 Z the map t : R ! R given by t(x) = x+n is a covering transformation for the covering p(x) = e2πix where p : R ! S1. The covering transformation for a given space forms a group which is called the group of covering transformations. Example, for the triple (R; S1; p) where p : R ! S1 given by p(x) = e2πix is a covering, the group covering transformation is Z. Definition 2.12. Let E; B be manifolds, E0 a submanifold of E; B0 a submanifold of B, and p : E ! B a continuous map. The quintet (E; E0; B; B0; p) is a branched covering if 0 0 (i) P jEnE0 : E n E ! B n B is a covering map; 0 0 (ii)P jE0 : E ! B is a covering map. Here B0 is called the branch locus and E0 is called the ramification locus. Example The quintet (C; 0; C; 0; f) , where f(z) = zn, is a branched covering. 2.3 Homotopy and Isotopy on Surfaces Isotpies are always homotopies. But in case of surfaces, we have more. Here, there is a the homotopy classes of closed simple curves correspond to the isotopy classes of simple closed curves. Definition 2.13. Let F be a surface and let C = c1; :::; cn be a collection of simple closed curves in F that have been isotoped to intersect in a minimal number of points. We say that C is filling if F n (c1 [ ::: [ cn) is a union of disks. Lemma 2.14 (Alexander Trick). Suppose that f : Dn ! Dn is a homeomorphism such that f j@Dn is the identity. Then f is isotopic to the identity. Proof. Define H : Dn × I ! Dn as follows 3 ( tf(x=t) if 0 ≤ kxk < t; H(x; t) = . x if t ≤ kxk ≤ 1 The schematic for the Alexander Trick is shown in the figure below. This is the required isotopy. Figure 4: Schematic for the above proof Theorem 2.15. Let F be a closed orientable surface and h : F ! F a homeomorphism. Now, if h is homotopic to the identity, then h is isotopic to the identity. 2.4 The Mapping Class Group The set of self-diffeomorphisms forms a group which is denoted by Diffeo(S) if S is oriented surface. Also, the set of orientation preserving self-diffeomorphism Diffeo+(S) and the set of self-diffeomorphisms homotopic to identity Diffeo0(S) form normal subgroups. Homeo- morphisms are isotopic to diffeomorphisms in case of surfaces, hence we will be considering only diffeomorphisms. Definition 2.16. Let S be a compact orientable surface. The mapping class group of S, denoted by MCG(S) is Diffeo+(S)= Diffeo0(S), the group of orientation-preserving self- diffeomorphisms of S modulo the subgroup consisting of self-diffeomorphisms of S homo- topic to the identity. 2 ∼ Lemma 2.17. MCG(T ) = SL(2; Z): We will denote the embedding of a closed regular neighborhood of a closed simple orientation preserving curve c as N(c) and an open regular neighborhood as η(c). Definition 2.18. Let c be an orientation-preserving simple closed curve in a compact surface S and let N(c) be a regular neighborhood of c that is oriented via the parametrization i : S1 × [O; 1] ! η(c). A map from f : S ! S is called a left Dehn twist around c if (i) f jSnN(c) is the identity map; and (ii) f jN(c) is the map of the annulus given by 2iπθ 2iπ(θ+t) 2iπθ 1 f(e ; t) = (e ) 8(e ; t) 2 S × [0; 1]: Likewise, a map f : S ! S is called a right Dehn twist around c if 4 (i) f jSnN(c) is the identity map; and (ii) f jN(c) is the map of the annulus given by 2iπθ 2iπ(θ−t) 2iπθ 1 f(e ; t) = (e ) 8(e ; t) 2 S × [0; 1]: Theorem 2.19 (Dehn, 1938; Lickorish, 1962). Every surface isomorphism is isotopic to a composition of Dehn twists i.e. the mapping class group is generated by Dehn twists. We will state two lemmas and use them to prove the above theorem. Lemma 2.20. If the simple closed curves α; β in the compact surface S intersect exactly once, then there is a pair f1; f2 of Dehn twists such that f2 · f1(α) is isotopic to β. Lemma 2.21. If α; β are oriented simple closed curves in the orientable surface S, then there is a series of Dehn twists f1; f2; :::; fk such that fk ·:::·f2 ·f1(α) is either disjoint from β or intersects β in exactly two oppositely oriented points. Sketch of the proof. Let S be a closed orientable surface a1; b1:::; ag; bg be collection of curves which cut S into a disk. Self-homeomorphisms take this collection of curves to another such collection of curves. Now, using a series of Dehn twists we reverse the effect of the surface diffeomorphism on the new collection of curves. This can be done because of the above stated lemmas. The collection of Dehn twists can be seen as "undoing" the effect of the surface diffeomorphism on the specified collection of curves. Using the Alexander Trick, we get that the map on the complementary disk is isotopic to identity.
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