GEOMETRY AND TOPOLOGY. Contents

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1 GEOMETRY AND TOPOLOGY NIMA MOSHAYEDI Notes, February 1, 2015 Contents Part 1. Topology 2 1. Topological spaces 2 2. Tools to construct topological spaces Cell complexes 8 3. Connectedness Separation Axioms 9 4. Compactness Homotopy The Fundamental group Functionality of the Fundamental group Coverings and the Fundamental group of the Circle The Seifert- van Kampen theorem Generators and Relations Compact surfaces The Fundamental group of RP n Construction of new Surfaces from known ones: Connected Sums Polygonal representation of surfaces Strategy of the Proof of the Classification Theorem The Euler characteristic 26 Part 2. Geometry Curves Curves in Euclidean Space (3 Dimension) Smooth Surfaces in R The First Fundamental Form Angles between Curves Areas isometric surfaces The Second Fundamental Form The Gaussian curvature The Gauss-Bonnet theorem Morse Functions Geodesics 48 Date: February 1,

2 2 N. MOSHAYEDI Part 1. Topology 1. Topological spaces Definition 1.1. Metric Space A metric on a set X is a function d : X X [0, ), (x, y) d(x, y) such that for all x, y, z X: (i) d(x, y) = 0 x = y (ii) d(x, y) = d(y, x) (iii) d(x, z) d(x, y) + d(y, z) (X, d) is called a metric space. Definition 1.2. Continuity Let f : X Y be a function between metric spaces (X, d X ) and (Y, d Y ). f is continuous at x X if for all ɛ > 0, there exists a δ > 0 such that d X (x, x ) < δ = d Y (f(x), f(x )) < ɛ for all x X. More formally, ɛ > 0, δ > 0 s.t. f(b X δ (x)) BY ɛ (f(x)). Definition 1.3. Neighborhood U is called a neighborhood of x if there exists an ɛ > 0 such that B ɛ (x) U. Proposition 1. Let f : X Y be a function between metric spaces and let x X. Then f is continuous if and only if for any neighborhood V of f(x), there exists a neighborhood U of x such that f(u) V. Definition 1.4. Open A subset O of a metric space (X, d) is open if it is a neighborhood of each of its points. Proposition 2. Let f : X Y be a function between metric spaces. Then f is continuous if and only if for any open subset O Y, f 1 (O ) is open. In words we say, f is continuous if and only if the reciprocal image of any open set is open.

3 GEOMETRY AND TOPOLOGY 3 Definition 1.5. Topology Let X be a set. A set O of subsets of X is a topology on X if the following holds. (i), X O (ii) Any union of elements of O is in O. O i O, i I = i I O i O. (iii) Any finite intersection of elements of O belongs to O. O 1,..., O n O = n i=1 O i O. Definition 1.6. Topological Space A topological space is a pair (X, O), where X is a set and O a topology on X. The sets O O are called the open sets. A subset A X is called closed set if and only if (X \ A) O. The closed sets are exactly the complements of the open sets. Definition 1.7. Zariski Topology The Zariski topology on R, C is defined such that the closed subsets are the zero loci of polynomials. Hence the open sets are, (R, C), (R, C) \ F, F (R, C) is a finite subset. Definition 1.8. Basis Let (X, O) be a topological space. B O is a basis for the topology O if and only if any open set O O is the union of elements of B. Lemma 1. Let X be a set and B P(X) a set of subsets of X s.t. X = U B U. Then the following are equivalent (1) There is a topology on X with basis B. (2) If U, V B, U V, x U V, then W B s.t. x W U V. Definition 1.9. Characterization of interior, closure and boundary Let (X, O) be a topological space, x X and A X a subset. (1) V X is a neighborhood of x if U O such that x U V. (2) x is an inner point of A if and only if A is a neighborhood of x. (3) The set Å of inner points is called the interior of A (4) x is a point of closure of A if and only if each neighborhood of x intersect A non-trivially. The set Ā of all the points of closure is called the closure of A. A point in Ā \ A is a limit point of A. (5) The points of closure of A that are not inner points are boundary points. The set A of boundary points of A is the boundary of A. Lemma 2. Let (X, O) be a topological space and let A X be a subset. (1) Å is open. Å = A A is open. (2) Ā is closed. Ā = A A is closed. (3) Ā = Å A. (4) A = Ā X \ A (5) X \ A = (X \ A) o, Ā = X \ (X \ A) o.

4 4 N. MOSHAYEDI Lemma 3. The following hold. (1) Ā is the smallest closed set containing A. Å is the largest open set contained in A. (2) =, X = X, Ā = Ā, A B = Ā B (3) =, X = X, Å = Å, (A B)o = Å B. Definition Homeomorphism f is called a homeomorphism if and only if it is bijective, continuous and f 1 is continuous. Definition Finer and Coarser Topologies Let O 1, O 2 be two topologies. O 1 is finer than O 2 if O 1 O 2, i.e. every set in O 2 is open in O 1. Then O 2 is coarser than O 1. This gives a partial order on topologies. Lemma 4. The following are equivalent: (1) O 1 is finer than O 2. (2) id : (X, O 1 ) (X, O 2 ) is continuous. (3) id : (X, O 2 ) (X, O 1 ) is open. 2. Tools to construct topological spaces Definition 2.1. Subspace Topology Let (X, O) be a topological space and A X a subset of X. O A = {O A O O} is called the subspace topology on A. (A, O A ) is called a subspace of X. Proposition 3. Let (X, O) be a topological space and A X. (1) The inclusion ι A : (A, O A ) (X, O) is continuous. (2) O A is the coarsest topology on A such that ι A is continuous. Proposition 4. Let f : X Y and X = n i=1 A i with A i closed for all 1 i n. Then f is continuous if and only if f i := f Ai : A i Y are continuous for all 1 i n. Definition 2.2. Product Topology I Let X, Y be two topological spaces and let X Y be the product space. We declare W X Y to be open if and only if for all (x, y) W, there exists U X, V Y open such that (x, y) U V W. The topology defined on this way is the product topology. Lemma 5. The product topology is the coarsest such that the projections X Y X and X Y Y are continuous.

5 GEOMETRY AND TOPOLOGY 5 Definition 2.3. Product Topology II Let {X i } i I be a set of topological spaces and let i I X i be their product as sets. Let O p be the set of subsets of i I X i that can be obtained by unions and finite intersection of the sets p 1 i (U), for U open in X (p j : i I X i X j projection). Then O p is the product topology on i I X i. Remark 2.1. O p is not the topology generated by products of open sets! The latter is the box topology, which is finer. The product topology and the box topology coincide when I is a finite index set. Definition 2.4. Disjoint Union (Topological sense) Let X, Y be two sets and define the disjoint union X Y = X {0} Y {1}. Let (X, O X ) and (Y, O Y ) be topological spaces. The disjoint union (topological sense) is the topological space (X Y, O) where O = {U V U O X, V O Y }. Lemma 6. The disjoint union topological is the finest on such that the inclusions and X X Y are continuous. Y X Y Definition 2.5. Quotient Topology Let (X, O) be a topological space, an equivalence relation on X and π : X X/, x [x] the projection onto the equivalence class. The quotient topology is defined by: O X/ is open π 1 (O) is open in X Lemma 7. The following hold. (1) π is continuous. (2) Let f : X/ Y. Then f is continuous iff f := f π is continuous. X f Y π f X/ Lemma 8. The quotient topology is the finest such that π is continuous. Example 2.1. We have the following examples:

6 6 N. MOSHAYEDI (1) Collapsing of a subspace to a point Let X be a topological space and A X. Let X/A = X/, where x y x = y or x, y A. Now let X = [0, 1] and A = {0, 1} X. Then X/A = S 1. For X = B n and A = B n S n 2 we get X/A = S n. (2) Cone of a space and suspension Let X be topology space. CX = (X [0, 1])/(X {0}) is called the cone of X. ΣX = (X [0, 1])/(X {0} X {1}) is called the suspension of X. The operation Σ might seem random, but it is of fundamental importance in algebraic topology. (3) Gluing at a point Let X, Y be topological spaces, x 0 X and y 0 Y. The spaces (X, x 0 ), (Y, y 0 ) are pointed topological spaces. The glued topological space is given by (X, x 0 ) (Y, y 0 ) = ((X Y )/({x 0 } {y 0 }), x 0 y 0 )

7 (4) Quotient by a group action GEOMETRY AND TOPOLOGY 7 Let G be a group acting continuously on a topological space X, i.e. G X X, (g, x) g x with e x = x, (g 1 g 2 ) x = g 1 (g 2 x) g 1 g 2 G, x X and g : X X is continuous for all g G. The orbit of x is Gx = {g x g G}. Define x y g G s.t. g x = y, i.e. y Gx. Then X/G := X/ is the orbit space. More over π : X X/G, x [x] π 1 ([x]) = Gx. We have the following examples: (a) X = S n, G = Z 2 = {±1}, 1 x = x, ( 1) x = x. In S n /Z 2, one identifies the antipode of S n. Hence S n /Z 2 = RP n, the real projective space. (b) X = R, G = Z n. g x = x+g action by integer translation. X/G = R n /Z n = T n, the n- dimensional torus. (5) Gluing of topological spaces Let X, Y be topological spaces, A X and ϕ : A Y continuous. Define X ϕ Y = X Y/ ϕ, where a ϕ b a = b or a = ϕ(b). Example 2.2. We have the following examples: (a) X = [0, 1], Y = R. A = {0, 1}, ϕ(0) = 0 and ϕ(1) = 1 (gluing a 1-cell). (b) Let D 2 := {x R 2 x 1} be the two dimensional unit disc and S 1 = D 2 = {x R 2 x = 1} be the unit circle. Let Y be a topological space and

8 8 N. MOSHAYEDI ϕ : S 1 Y be continuous. The quotient is the gluing of a 2-cell. We can also go on with higher dimensional cells. This is a very general method to construct topological spaces Cell complexes. Let S 0 be a set of points (discrete space) Glue a 1-ball B 1 to S 0 via ϕ 1 1 : B 1 S 0 Repeat as many times as needed, with maps {ϕ 1 n : B 1 S 0 } We get the topological space S 1, the 1.scelleton Glue a 2-ball B 2 to S 1 via ϕ 2 1 : B 2 S 1 Repeat as many times as needed, with maps {ϕ 2 n : B 1 S 0 } We get the topological space S 2, the 2-scelleton Continue by attaching n-balls B n to S n 1 via ϕ n k : Bn S n Connectedness Definition 3.1. Connected Space A topological space (X, O) is called connected if X cannot be decomposed into the union of two disjoint non-empty open subspaces, i.e. X = O 1 O 2, O 1, O 2 O, O 1 O 2 = O 1 O 2 for X connected. Lemma 9. (X, O) is connected iff and X are the only subspaces of X that are both open and closed. Example 3.1. We have the following examples: (1) (a, b) (c, d), a < b < c < d is disconnected. (2) (a, b) R is connected for a < b. Indeed, assume U, V open with U V = (a, b) and U V = with U, V non-empty. Let u U, v V and assume u v. Let S = {s (a, b) (u, 1) U} and s o = sup S (a, b), s 0 v. Then s 0 U because U is open and s 0 V because V is open. This is a contradiction, which proves the claim. Proposition 5. Let X be a topological space and A X connected (in the induced topology). Then A B Ā = B is connected.

9 GEOMETRY AND TOPOLOGY 9 Proposition 6. Let f : X Y be continuous. Then if X is connected = f(x) is connected. Proposition 7. Intermediate Value Theorem Let X be a connected space, f : X R continuous and t, s f(x). Then f takes every values between t and s. Definition 3.2. Path-connected Space X is called path-connected if for each x, y X, there exists a continuous function ϕ : [0, 1] X with ϕ(0) = x and ϕ(1) = y. We say that ϕ is a path from x to y. Proposition 8. If X is path-connected then X is connected. Remark 3.1. There are spaces that are connected but not path-connected. For example let f : (0, ) R such that f(x) = sin ( 1 x). Let X be a space given by X = {(x, f(x)) x (0, )} {{0} [0, 1]} R with the topology induced from the metric topology on R 2. Then X is connected but not path-connected Separation Axioms. Definition 3.3. T1-space (1) X is called a T 1-space if for all x, y X with x y there exists neighborhoods U x of x and U y of y such that x U y and y U y. (2) X is called a T 2-space, or a Hausdorff space if for all x, y X with x y, there exists neighborhoods U x of x and U y of y such that U x U y =. Any T 2-space is also a T 1-space. Example 3.2. We have the following examples: (1) R with the metric topology is T 2. (2) R with the strange topology in which the only open sets are of the form (, a) is neither T 2 nor T 1. (3) R with the Zariski topology is T 1 but not T 2. Lemma 10. A metric space with the induced topology is Hausdorff. Definition 3.4. Let X be a topological space and let {x n } n N be a sequence of points of X. Then x is a limit of {x n } n N iff for each neighborhood U of x, there exists n U N such that x n U for each each n n U. Lemma 11. In a Hausdorff space any sequence has at most one limit point.

10 10 N. MOSHAYEDI Definition 3.5. Metrizable Space A topological space (X, O) is metrizable, when there exists a metric on X that induces O. If (X, O) is metrizable, then it is T 2. The Zariski topology on R is not metrizable. 4. Compactness Definition 4.1. Compact Space A topological space (X, O) is called (1) quasi-compact if every cover of X by open sets admits a finite subcover. In general X = i I U i, U i O = i 1,..., i n I s.t. X = U i1 U in (2) compact if X is quasi compact and Hausdorff. Example 4.1. We have the following examples: (1) R is quasi-compact in the Zariski topology. (2) The closed cube in R n with the metric topologyy is compact. Indeed, let Q 1 = [a 1, b 1 ] [a n, b n ] R n, where a = diam(q 1 ) = ( n i=1 (b i a i ) 2 ) 1/2 = sup x,y Q1 d(x, y). By contradiction: Assume that Q 1 i I U i with no finite subcover. Split Q 1 into 2 n smaller cubes Q 1 1,..., Q 2n 1 by splitting each intervals into 2 equal pieces. Then there is at least one of the small cubes Q 2 {Q 1 1,..., Q 2n m 1 } that is not covered by any subcover of {U i } i I. Let diam(q 2 ) = a. Now iterate the procedure. Then we get 2 Q 1 Q 2 Q 3 with diam(q i ) = a for i N. None of the Q 2 i k are covered by a finite subcover of {U i } i I. As diam(q k ) k 0, there exists a unique x 0 R n such that x 0 k=1 Q k. Then x 0 U i0 for at least one i o I. As U i0 is open, x 0 B ɛ (x 0 ) U i0 for ɛ > 0 small enough. Hence there is a k such that Q k U i0. But this is a contradiction and therefore Q 1 is compact. Proposition 9. Let X be quasi-compact and f : X Y continuous. Then f(x) is quasi-compact. If Y is Hausdorff, then f(x) is compact. Proposition 10. The following hold. (1) Let X be quasi-compact and A X a closed subset. Then A is quasi compact. (2) Let X be Hausdorff and A X a compact subset. Then A is closed. Proposition 11. Let (X, d) be a metric space and let A X be a compact subset. Then A is bounded in the sense that r > 0 and x X such that A B r (x). Proposition 12. Let (X, d) be a metric space. Then we have the following implication for a subset A X: A is compact = A is bounded and closed. But not the reverse!

11 GEOMETRY AND TOPOLOGY 11 Theorem 12. Heine-Borel In R n, the compact subsets are the closed and bounded subsets. Proposition 13. Let X be quasi-compact, Y Hausdorff and f : X Y continuous and bijective. Then f is a homeomorphism, i.e. f 1 is continuous. Theorem 13. Bolzano-Weierstrass Let (X, d) be a metric space and let X be compact. Then every sequence in X has a convergent subsequence. Theorem 14. Tychonoff Let X be the product space X = λ Λ X λ with the product topology. Then X is compact iff each X λ is compact. 5. Homotopy Definition 5.1. Homotopy and Homotopic Functions Let X, Y be two topological spaces and f, g : X Y be continuous functions. Then f is homotopic to g (written f g) if there exists a continuous function H : X [0, 1] Y such that: (1) H(x, 0) = f(x) for all x X (2) H(x, 1) = g(x) for all x X We say H is a homotopy from f to g. Remark 5.1. H defines a continuous family h t for t [0, 1], of functions from X to Y, h t (x) = H(x, t) sucht that h 0 = f and h 1 = g. We say f can be continuously deformed into g. Example 5.1. We have the following examples: (1) Let f, g : X R n be continuous functions. Then f g, because H : X [0, 1] R n with H(x, t) = (1 t)f(x) + tg(x) is continuous and H(x, 0) = f(x), H(x, 1) = g(x). (2) Let f, g : S 1 R 2 \ {0}. We will show that f is not homotopic to g. Lemma 15. Let X, Y be two topological spaces. Then is an equivalence relation on the set of continuous function from X to Y. Definition 5.2. Homotopy Class The equivalence class [f], with respect to, of a continuous function f : X Y is called the homotopy class of f. Lemma 16. Let f, f : X Y and g, g : Y Z be continuous functions sucht that f f and g g. Then g f g f.

12 12 N. MOSHAYEDI Definition 5.3. Homotopy Equivalence (1) f : X Y is called a homotopy equivalence if there exists a continuous function g : Y X such that g f id X and f g id Y. (2) If f is a homotopy equivalence, then X and Y are called homotopic equivalent, X Y. (3) X is said to be contractible if X, where is a point. Example 5.2. We have the following examples: (1) Let X = S 1 and Y = R 2 \ {0}. Then S R 2 \ {0}. Indeed, see S 1 R 2 \ {0}. Take the inclusion f : S 1 R 2 \ {0}, x x and the radial projection g : R 2 \ {0} S 1, y y. Then g f = id y S 1. We need to show that f g is homotopic to id R 2 \{0}. Take H : R 2 \ {0} [0, 1] R 2 \ {0}, (y, t) (1 t) y y + ty. Then H(y, 0) =, y y H(y, 1) = y. Therefore f g id R 2 \{0} and hence S 1 R 2 \ {0}. (2) R n is contractible. Indeed, take f : R n {0}, x 0 and g : {0} R n, 0 0. Then g f : R n R n, x 0 and f g = id {0}. Now H : R n [0, 1] R n, (x, t) tx is a homotopy between g f and id R n. Definition 5.4. Relative homotopic Let X, Y be two topological spaces and A X a subspace. Let f, g : X Y be continuous functions such that f A = g A. f is called homotopic to g relative to A (written f g rel A) if there exists a homotopy H from f to g that is constant on A, i.e. H : X [0, 1] Y, H(x, 0) = f(x), H(x, 1) = g(x), H(a, t) = f(a) = g(a) for all a A. Remark 5.2. When A =, we recover the usual notion of homotopy. Definition 5.5. Deformation retract Let X be a topological space and A X. A is called a deformation retract of X if there exists a continuous function f : X X such that f(x) A, f A = id and f id X rel A. Remark 5.3. Notice that following hold. f is a homotopy equivalence between X and A relative to A. If A is a deformation retract of X, one can shrink X continuously onto A, keeping the points of A fixed.

13 GEOMETRY AND TOPOLOGY The Fundamental group Definition 6.1. Homotopy of Paths Let X be a topological space. A path in X is a continuous function f : [0, 1] X. A homotopy of paths (with fixed endpoints) between paths p 1, p 2 : [0, 1] X is a homotopy between p 1 and p 2 relative to {0, 1}, i.e. a map F : [0, 1] [0, 1] X such that f t (x) = F (x, t), f t (0) = x 0, f t (1) = x 1, f 0 (x) = p 1 and f 1 (x) = p 2. The homotopy of paths defines an equivalence relation on the space of paths from x 0 to x 1. Definition 6.2. Loop with Base point A loop based at x 0 X is a path starting and ending at x 0, i.e. a continuous map f : [0, 1] X such that f(0) = f(1) = x 0. We write π 1 (X, x 0 ) = {[f] f : [0, 1] X, f(0) = f(1) = x 0 } for the set of homotopy classes of loops based at x 0. We call x 0 the base point. Definition 6.3. Concatination Let f, g : [0, 1] X be two paths in X such that f(1) = g(0). We can define the concatenation f g of f and g as the path { f(2t), t [0, 1 f g(t) = ] 2 g(2t 1) t [ 1, 1] 2 Corollary 1. The concatenation of homotopy classes of paths is well defined, i.e. f 0 f 1 and g 0 g 1 = f 0 g 0 f 1 g 1. Therefore [f] [g] := [f g] is well defined. if Lemma 17. π 1 (X, x 0 ) is a group with respect to the concatenation. Definition 6.4. π 1 (X, x 0 ) is called the fundamental group of X. Lemma 18. Auxiliary Lemma The reparameterization of a loop does not change its homotopy class. Lemma 19. There is a group isomorphism β h : π 1 (X, x 1 ) π 1 (X, x 2 ) between π 1 (X, x 1 ) and π 1 (X, x 2 ) defined by β h ([f]) = [h (f h)].

14 14 N. MOSHAYEDI Remark 6.1. Notice that the following hold. If X is not path-connected there is no path from x 0 to x 1. Then we can have π 1 (X, x 0 ) π 1 (X, x 1 ). If X is path-connected, we can speak of the fundamental group π 1 (X) without reference to a base point. Definition 6.5. Simply Connected Space X is said to be a simply connected topological space if X is path-connected and the fundamental group is trivial. There are two important facts: (i) If X is simply connected, any loop in X can be contracted continuously to a point. (ii) A contractible space is simply connected. Lemma 20. Let F : [0, 1] [0, 1] X be a continuous function. Let α(t) = F (0, t), β(t) = (1, t), γ(s) = F (s, 0) and δ(1) = F (s, 1). Then δ ᾱγβ Functionality of the Fundamental group. Let f : X Y be a continuous function between two topological spaces X and Y. Let x 0 X and y 0 = f(x 0 ). Given a path p : [0, 1] X, we can push it forward using f as follows. f p : [0, 1] Y is a path in Y defined by f p(t) = f(p(t)). If P : [0, 1] [0, 1] X is a path homotopy between p 0 and p 1, then f P is a path homotopy between f p 0 and f p 1. Therefore f passes to a well defined function on homotopy classes of paths in X. We have a well defined map f : π 1 (X, x 0 ) π 1 (Y, y 0 = f(x 0 )), [σ] f [σ] = [f σ] Proposition 14. Homeomorphic spaces have identical fundamental groups. If f is a homeomorphism, then f induces an isomorphism on the space of loops based at x 0 and y 0. It also maps bijectively the homotopies between loops. Hence f is an isomorphism between π 1 (X, x 0 ) and π 1 (Y, y 0 ). The basic problem in topology is to classify topological spaces up to homeomorphisms. The fundamental group is useful for this purpose because spaces admitting different fundamental groups are distinct up to homeomorphisms. Moreover π 1 is an example of a topological invariant. Proposition 15. If φ : X Y is a homotopy equivalence between two topological spaces X and Y, then φ : π 1 (X, x 0 ) π 1 (Y, φ(x 0 )) is an isomorphism. Therefore homotopy equivalent spaces have isometric fundamental groups.

15 GEOMETRY AND TOPOLOGY Coverings and the Fundamental group of the Circle Definition 7.1. Covering Map Let E and X be two topological spaces. A continuous function p : E X is called a covering map iff for all x X, there exists a neighborhood U of x such that p 1 (U) = β V β with p Vβ : V β U being homeomorphisms and the V β are disjoint for all β. Such a neighborhood will be called admissible. p 1 (x) is a disjoint set, called the fiber of the covering over x. If the fiber is a finite set of k points, the covering is called a k-sheeted covering. Example 7.1. We have the following examples: (1) Let p : R S 1 be a map such that t (cos 2πt, sin 2πt). Then p 1 (x) = Z for x S 1. (2) Let p n : S 1 S 1 be a map such that (cos 2πt, sin 2πt) (cos 2πnt, sin 2πnt). Then p 1 (x) = {1,..., n} for x S 1. (3)

16 16 N. MOSHAYEDI Definition 7.2. Equivalent Coverings Two coverings p 1 : E 1 X and p 2 : E 2 X are called equivalent if there is a homeomorphism f : E 1 E 2 satisfying p 2 f = p 1. Proposition 16. Lifting Lemma Let X, E and Y be topological spaces. Let p : E X be a covering map. Let σ : Y [0, 1] X be a continuous family of paths in X parameterized by Y, such that σ(y, ) = f(y) for a continuous function f : Y X. Assume that f is lifted to a function f : Y E such that p f = f. Then there exists a unique lift σ : Y [0, 1] E such that p σ = σ ( σ(y, 0) = f(y)). Remark 7.1. The proposition sais that if we fix x 0 p 1 (x 0 ), then there is a unique path σ in E lifting σ and starting at x 0. σ is called the lift of σ starting at x 0. Proposition 17. Path Homotopy Lifting Let X and E be two topological spaces. Let p : E X be a covering map. Let σ 0 and σ 1 be paths in X with x 0 = σ 0 (0) = σ 1 (0), σ 0 (1) = σ 1 (1) and σ 0 σ 1 rel {0, 1}. Let x 0 p 1 (x 0 ) and σ 0, σ 1 be the lifts of σ 0, σ 1 respectively, starting at x 0. Then σ 0 σ 1 rel {0, 1}. Corollary 2. Let σ 0, σ 1 be two paths in X such that x 0 = σ 0 (0) = σ(0), σ 0 (1) = σ 1 (1) and σ 0, σ 1 be their lifts at x 0 p 1 (x 0 ). If σ 1 (1) σ 0 (1), then σ 0 and σ 1 are not path homotopic. Hence the endpoint of the lifts can give information about the homotopy class of a path. Proposition 18. π 1 (S 1, 1) Z Proposition 19. Brower s Fixed point Theorem for the Disc Let D 2 = {x R 2 x 1} and f : D 2 D 2 be continuous. Then there is a fixed point of f, i.e. a point x D 2 such that f(x) = x.

17 GEOMETRY AND TOPOLOGY 17 Proposition 20. Borsuk-Ulam Let f : S 2 R 2 be continuous. Then there exists x S 2 such that f(x) = f( x). This also holds for f : S n R n. Corollary 3. Let S 2 A 1 A 2 A 3 and A i closed for i {1, 2, 3}. Then there is at least one A i containing a pair of antipodal points, i.e. there exists x S 2 with x, x A i. 8. The Seifert- van Kampen theorem We want to find a tool to compute the fundamental group of a topological space. Definition 8.1. Free product The free product α G α of the groups G α is defined as follows. As a set, α G α is the set of finite words g 1 g m for m 0, where g i G αi and such that α i α i+1 (such words are called reduced.) Equivalently one can allow arbitrary words, but make identification g 1 g i g i+1 g m = g 1 g g n whenever g i, g i+1 G α, g i g i+1 = g. The product in α G α is given by the concatination of words (possibly followed by reduction): (g 1 g m )(h 1 h n ) = g 1 g m h 1 h n. The right hand side can be reduced if g m and h 1 belong to the same group and the process goes further if g m = h 1 1. The neutral element is the empty word, written 1. The inverse of g 1 g m is given by gm 1 g1 1. Example 8.1. Z }{{} {a n n Z} Z }{{} {b n n Z} (i) a 2 b 2 ab 2 Z Z (ii) (a 2 b 1 ab 2 )(b 2 a) = a 2 b 2 ab 4 a (iii) a 2 b 2 ab 2 (b 2 a 1 ) = a 2 b 2 Theorem 21. Universal property of the Free product Let ϕ α : G α H be a group homomorphism. Then there exists a unique ϕ : α G α H such that ϕ(g 1 g n ) = ϕ α1 (g 1 ) ϕ αn (g n ), g i G αi. For α {1, 2}, we get the following commutative diagram: G 1 G 1 G 2 G 2 ϕ1 ϕ. H ϕ 2

18 18 N. MOSHAYEDI 8.1. Generators and Relations. A useful way to describe many groups is as follows. Z Z =: F n is the free group on n generators. Its elements are all reduced words in the generators. Pick words w 1,..., w k. Elements of the form gw i g 1, for g F n, generate a normal subgroup N of F n. We write F n /N = g 1,..., g n w 1 = 1,..., w k = 1 with g 1,..., g n the generators of F n. The elements of F n /N are all the words in the generators, subject to the relation w i = 1. F n /N is said to admit a presentation with generators and relations. Example 8.2. Z 2 = g g 2 = 1 Z Z = g 1, g 2 g 1 g 2 = g 2 g 1 Definition 8.2. Amalgamated product Let A, G 1, G 2 be groups and ψ 1 : A G 1, ψ 2 : A G 2 be group homomorphisms. Let N be the normal subgroup of G 1 G 2 generated by all elements of the form ψ 1 (a)(ψ 2 (a)) 1. Then the free product of G 1 and G 2 with amalgamated subgroup A is given by G a A G 2 = G 1 G 2 N. G 1 A G 2 is composed of the same words as G 1 G 2, in which the extra relation ψ 1 (a) = ψ 2 (a), a A has been imposed. Theorem 22. Universal property for the Amalgamated product A ψ 1 ψ2 G 1 i 1 G 1 A G 2 i 2 G 2 γ1 ϕ. H Whenever there is H, γ 1, γ 2 making the diagram commute,!ϕ : G 1 A G 2 H completing it. Example 8.3. Consider the case where a topological space X is the union of subspaces. Let X = U 1 U 2, where U 1, U 2 and U 1 U 2 are all path-connected. We pick x 0 U 1 U 2. Write γ i : U i X and k i : U 1 U 2 U i. Define also ι i : π 1 (U i, x 0 ) π 1 (X, x 0 ) where ι i = γ i and γ 2

19 GEOMETRY AND TOPOLOGY 19 ψ i : π 1 (U 1 U 2, x 0 ) π 1 (U i, x 0 ) where ψ i = k i. Therefore we get the following diagram: π 1 (U 1 U 2 ) ψ1 π 1 (U 1 ) ψ 2 ι 1 π 2 (U 2 ) ι π 1 (X) 2 Note that X is the universal object for the diagram. The Seifert-van Kampen theorem sais that π 1 (X) is the universal object for the diagram above. Theorem 23. Seifert-van Kampen If we have everything as in the example above, then π 1 (X) = π 1 (U 1 ) π1 (U 1 U 2 ) π 1 (U 2 ), π 1 (U 1 U 2 ) ψ 2 π 1 (U 2 ) ι 1 π 1 (X) ι 2 ψ1 π 1 (U 1 ) γ1 ϕ. H i.e. for any group H and γ i : π 1 (U i ) H such that γ 1 ψ 2 = γ 2 ψ 1,!ϕ : π 1 (X) H that makes the diagram commute. Remark 8.1. The Seifert-van Kampen theorem allows us to compute π 1 (X) from the knowledge of π 1 (U 1 ), π 1 (U 2 ), π 1 (U 1 U 2 ), ψ 1, and ψ 2. Example 8.4. Let X = S 1 S 1. Then π 1 (X) = π 1 (S 1 ) 0 π 1 (S 1 ) = Z Z. γ 2 9. Compact surfaces Definition 9.1. Manifold A topological space M is called a manifold if (1) M is Hausdorff (2) M has a countable basis of its topology (3) M is locally homeomorphic to R n for some n. I.e. for each point x M, there exists an open neighborhood of x that is homeomorphic to an open subset of R n. Remark 9.1. n is called the dimension of the manifold M. Example 9.1. We have the following examples:

20 20 N. MOSHAYEDI R n, S n, T n = S } 1 {{ S } 1 are manifolds of dimension n. n The wedge of 2 circles is not a manifold because the crossing point does not satisfy condition (3). Definition 9.2. Surface A surface is a 2-dimensional manifold. Example 9.2. R 2, S 2, T 2, RP 2 := S 2 / ± id, Klein bottle. Remark 9.2. The fundamental group of S n is π 1 (S n, x 0 ) = 0 (trivial) for all n 2. Proposition 21. Invariance of the Domain Let U R n, V R m be open subsets. Then there is no homeomorphism between U and V if n m. Remark 9.3. This implies that a manifold has a well defined dimension (on each connected component) The Fundamental group of RP n. RP n is the quotient of S 2 by the antipodal map. Use the parameterization S 1 [ π 2, π 2 ] S2, ( x, t) cos t( x, 0) + sin t( 0, 1). Consider also the antipodal map given by ( x, t) ( x, t). We have a covering p : S 2 RP 2. Take U + = p(open upper hemisphere) RP 2 and U 2 = p(s 1 ( π 4, π 4 )). Therefore we get U 1 U 2 Moebius-central circle = cylinder. We have π 1 (U 1 ) = 1, π 1 (U 2 ) = Z and π 1 (U 1 U 2 ) = Z. Moreover for a π 1 (U 1 U 2 ), ψ 2 (a) = 2a π 1 (U 2 ). Therefore π 1 (RP 2, x 0 ) = Z 2. Z 1 2 Z π 1 (RP 2 ) 9.2. Construction of new Surfaces from known ones: Connected Sums. Definition 9.3. Connected Sum Let F 1 and F 2 be two connected surfaces. The connected sum F 1 #F 2 is constructed as follows. (1) Delete an open disc in F 1 and one in F 2. (2) One obtains two surfaces with boundary, i.e. two topological spaces locally homeomorphicc to R 2 or {(x, y) R 2 x > 0} (bounderies). (3) The boundaries are a circle in each surface. Identify them with a homeomorphism. After the identification, one gets a closed surface, F 1 #F 2. Proposition 22. F 1 #F 2 is independent of (1) The choice of the open discs in F 1 and F 2, (2) The homeomorphism used to identify the boundary circles. Moreover # is associative: (F 1 #F 2 )#F 3 F 1 #(F 2 #F 3 ).

21 GEOMETRY AND TOPOLOGY 21 Example 9.3. T 2 T 2 #T 2 T 2 #T 2 #T 2 Theorem 24. Classification of Compact Closed Surfaces Any compact closed surface is homeomorphic to one of the following: (1) S 2 (2) A connected sum of k tori T 2, k 1. (3) A connected sum of k projective spaces RP 2, k 1. Moreover, these surfaces are all distinct up to homeomorphism Polygonal representation of surfaces. The following way of continuing surfaces is useful. Let P be a polygon with an even number of sides, say 2n. Orient and label edges with n labels, so that each label is used twice. Identify the edge carrying the same label by mean of homeomorphisms preserving the orientation of the edges. I.e. take the quotient topological space. This construction always yields surfaces.

22 22 N. MOSHAYEDI We call this construction a polygonal presentation of a surface. Let us check in one example that the quotient is locally homeomorphic to R 2. We see why we need to identify the edges 2 by 2. An unidentificated edge would look like = Not locally homeomorphic to R 2. 2 identified edges would look like The surfaces we are familiar with can all be presented by quotients of polygons: But the presentation is not unique! For instance The operation of taking the connected sum can easily be presented in terms of polygons: Notation: We will denote a polygonal presentation of a surface by the sequences of labels of its edges, going clockwise, with a () 1 representing the edges oriented counterclockwise. Defined only up to cyclic permutation. We call pairs of edges appearing as a a or a 1 a 1 straight and pairs appearing as a a 1 or a 1 a opposed.

23 GEOMETRY AND TOPOLOGY 23 Lemma 25. The connected sum of k tori can be presented by (a 1 b 1 a 1 1 b 1 1 )(a 2 b 2 a 1 2 b 1 2 ) (a k b k a 1 k b 1 k ) Lemma 26. The connected sums of k projective spaces can be presented by (a 1 a 1 )(a 2 a 2 ) (a k a k ) 9.4. Strategy of the Proof of the Classification Theorem. Do following steps: (1) Show that any compact, closed surface admits a polygonal presentation. (2) Show that any polygonal presentation is homeomorphic to one of the canonical presentation above. (3) Show that the surfaces corresponding to the canonical presentations are distinct up to homeomorphism (for this, use the fundamental group). We will not prove (1), but here is the idea: Let F be a compact and closed surface. As it s compact, it admits a finite covering by open discs. With some work, one can use the covering to show that F admits the structure of a CW-complex. Definition 9.4. CW-complex A CW-complex is a topological space computed inductively as follows: Start from a discrete set of points S 0, the 0-sceletton. Take a union of 1-balls (line segments) α B1 α and a continuous map f 1 : ( α B1 α) S 0, and glue along f 1 (i.e. quotient by x f 1 (x), x ( α B1 α)). This yields the 1-sceletton S 1. Take a union of n-balls α B1 α and a continuous map f n : ( α B1 α) S n 1 and glue along f n. The balls B n α are called the cells of dimension n (n-cells) of the complex. Example 9.4. Any polygonal presentation of a surface is a CW-complex. Theorem 27. Any surface admits a CW-complex structure. Given a CW-complex, triangulate the cells. Cut the triangulation so that it lays fat, keeping track of the identification

24 24 N. MOSHAYEDI Delete the inner edges = polygonal presentation of the surface We therefore assume that a surface F admits a polygonal presentation. 2) We now want to show that it is homeomorphic to one of the canonical presentation. We do this in several steps. A) Proposition 23. A polygonal presentation can always be replaced by an equivalent one that contains only one vertex. Equivalent means here that the associated surfaces are homeomorphic. B) Make the straight pairs of edges contiguous C) Pairs of labels a and b appearing as a b a 1 b 1 up to cyclic permutation are called crossed quadruplets. Proposition 24. The crossed quadruplets can be made contiguous aba 1 b 1 D) No uncrossed pairs. All the straight pairs and crossed quadruplets are contiguous. Can they be repeated by lonely opposed pairs? No! If there were such pairs left, without another pair turning it into a crossed quadruplet, then P can not be identified with Q in the quotient. No uncrossed pairs are left. E) If there are straight pairs, replace the crossed quadruplets by straight pairs.

25 GEOMETRY AND TOPOLOGY 25 2 new straight pairs and 1 less crossed quadruplet. Use B) to make the straight pairs contiguous. One can eliminate all crossed quadruplets if there is at least one straight pair. We showed that any polygonal presentation is equivalent to a canonical one. 3) We now show that the surfaces obtained from the canonical presentations are all distinct up to homeomorphisms. Idea: Compute their fundamental groups and show that they are all distinct. As the fundamental group is preserved by homeomorphisms, this will show 3). To compute the fundamental group of a surface given by a polygonal presentation, we proceed for the example of the torus above. We assume that there is only one vertex. π 1 (U 1 ) = Z Z Z Z Now with Seifert-van Kampen we get π 1 (U 2 ) = 1 π 1 (F ) = a, b, c, d a 2 bd 1 c 1 d 1 b 1 c = 1. This example above generalizes to any polygonal presentation with one vertex and we get the following theorem. Theorem 28. The fundamental group of a surface obtained from a polygonal presentation with labels a 1,..., a n and whose circumference forms the word w in the labels is π 1 (F ) = a 1,..., a n w = 1 Therefore the fundamental group of any surface is one of the following (1) π 1 (S 2 ) (2) π 1 (T 2 # #T }{{} 2 ) = a 1, b 1,..., a n, b n a 1 b 1 a 1 1 b 1 1 a n b n a 1 n n (3) π 1 (RP 2 # #RP }{{} 2 ) = a 1,..., a n a 2 1 a 2 n = 1 n Are any of the these groups isomorphic? b 1 n

26 26 N. MOSHAYEDI Given two group presentations, it is very hard to decide whether the describe isomorphic groups. In fact the problem is undecidable: no algorithm can solve correctly every instance of the problem. The trick is to abelianize them. Definition 9.5. Abelianization Given a group G admitting the presentation a 1,..., a n w 1 = 1,..., w k = 1, the abelianization G ab of G is the group given by the presentation a 1,..., a n w 1 = 1,..., w k = 1, a i a j = a j a i, i, j = 1,..., n i.e. it is the group admitting the same relation, but whose generators commute (hence G ab is abelian). Of course, if two groups have different abelianizations, they are different. We have π ab π ab 1 (S 2 ) = 1 1 (T 2 # #T }{{} 2 ) = a 1, b 1,..., a n, b n a i a j = a j a i, a i b j = b j a i, b i b j = b j b i Z } {{ Z } n 2n 1 (RP 2 # #RP }{{} 2 ) = a 1,..., a n a i a j = a j a i, (a 1 a n ) 2 = 1 Z } {{ Z } Z 2 n n 1 π ab These groups are all non-isomorphic. The fundamental groups are non-isomorphic. The canonical presentations yield distinct surfaces up to homeomorphism. 10. The Euler characteristic The fundamental group is a topological invariant, but it s not always easy to deal with. We define here a simpler topological invariant of surface. Observation: We notice the following: Tetrahedron: 4 vertices, 6 edges, 4 faces = = 2. Cube: 8 vertices, 12 edges, 6 faces = = 2. Octahedron: dual to the cube = = 2. Dodecahedron: 20 vertices, 30 edges, 12 faces = = 2. Icosahedron: dual to the dodecahedron = = 2

27 GEOMETRY AND TOPOLOGY 27 The platonic solids can be seen as CW-complex representation of the sphere. For any such CW-complex representation, we have χ(s 2 ) = #vertex #edges + #faces = 2 Definition Euler Characteristic Given a CW-complex with k n n-cells, the Euler characteristic χ is defined by χ = k 0 k 1 + k 2 k 3 ±... = ( 1) n k n, where we assume that the CW-complex is finite dimensional, i.e. that k n = 0 for n > N. Poperties of the Euler Characteristic: We have following properties: χ is a homotopy invariant, i.e. homotopy equivalent CW-complexes have the same Euler characteristic. χ is additive under disjoint union, i.e. χ(x Y ) = χ(x) + χ(y ). χ is multiplicative under products, i.e. χ(x Y ) = χ(x)χ(y ) If M is a k-sheeted covering of M, then χ( M) = k χ(m) Using the canonical presentations, we can easily compute the Euler characteristic of any surface. (1) χ(s 2 ) = 2, as we already saw. (2) χ((t 2 ) #n ) = 1 2n + 1 = 2n + 2. In particular χ(t 2 ) = 0 (3) χ((rp 2 ) #n ) = 1 n + 1 = n + 2. Remark Note following χ((t 2 ) #n ) = χ((rp 2 ) #2n ) despite the fact that (T 2 ) #n and (RP 2 ) #2n are not homotopy equivalent. The Euler characteristic is less refined than the fundamental group as a homotopy invariant. For oriented surfaces (i.e. (1) and (2)), one sometimes prefers the genus g, defined by g = 2 χ 2 The genus counts the number of holes in an oriented surface. n=0

28 28 N. MOSHAYEDI Part 2. Geometry 11. Curves In the following I denotes any non-empty interval in R, or R itself. Definition Parameterized Curve A parameterized curve is a C -differentiable function c : I R n. We write a dot for the tangent map c 1 (t) c 1(t) c(t) =., ċ(t) =. c n (t) c n(t) Recall that C -differnetiable (or C for short) means that the derivative of any order exist. Definition Regular Curve A curve c is called regular if ċ(t) 0 for all t I. Definition Change of Parameterization and Reparameterization Let c : I 1 R n be a parameterized curve. A change of parameterization is a bijective function ϕ : I 2 I 1, for I 2 some interval on R, such that ϕ and ϕ 1 are C. The curve c = c ϕ is also a parameterized curve and is called the reparameterization of c by ϕ. Lemma 29. If c is regular, then c is regular. Definition Orientation-preserving and Orientation-reversing The change of parameterization ϕ is called orientation-preserving if ϕ (t) > 0 t I 2, and orientation-reversing if ϕ (t) < 0 t I 2. Definition Length of a Regular Curve Let c : I R n be a regular parameterized curve. Then L(c) = ċ(t) dt [0, ] is the length of c. A curve with finite length is called rectifiable. I Lemma 30. If ϕ : I 2 I 1 is a change of parameterization, then L(c ϕ) = L(c). Definition Parameterization by arc-length A curve c is parameterized by arc-length iff ċ(t) = 1 for all t I. Lemma 31. Let c : I R n be a regular curve. Then one can reparameterize c by arc-length.

29 GEOMETRY AND TOPOLOGY 29 Definition Plane Curve A plane curve is a curve whose target is R 2. Definition Curvature For a curve c, not necessarily parameterized by arc-length, the curvature is defined as where J = ( ) K c (t) = 1 Jċ(t), c(t), ċ(t) 3 Lemma 32. Let F : R 2 R 2 be an orientation-preserving isometry and c : I R 2 a curve. Then K F c (t) = K c (t). Lemma 33. Let c : [a, b] R 2 be a curve parameterized by arc-length. Then there exists a C -function θ : [a, b] R such that ( ) cos θ(t) ċ(t) =, sin θ(t) where θ is defined up to addition of 2πk for k Z. Definition Periodic Curve A periodic curve is a curve c : R R 2 such that c(t + L) = c(t) for all t R and L > 0 is the smallest number such that this is true. Definition Rotation Index Let c be a periodic curve and θ as in the Lemma 33. Then the expression n c := 1 (θ(l) θ(0)) Z 2π is called the rotation index of c.

30 30 N. MOSHAYEDI Remark As ċ(t + L) = ċ(t), we get for all t R. ( ) cos θ(t + L) sin θ(t + L) = n c = 1 (θ(t + L) θ(t)) Z 2π ( ) cos θ(t) and sin θ(t) Theorem 34. Let c be a periodic curve with period L and let K c : R R be its curvature. Then n c = 1 2π L 0 K c (t)dt. Remark Notice that the following hold: n c is invariant under (orientation-preserving) homeomorphisms of R 2, which means that n c is a topological invariant. K c is only invariant under the isometries of R 2, which means that K c is only a geometrical invariant. Theorem 34. states that nevertheless, L 0 K c (t)dt is invariant under homeomorphism of R 2. Definition Simple Curve A periodic curve is simple iff it does not cross itself, i.e. iff c(t 1 ) = c(t 2 ) = t 2 t 1 kl for k Z and L the period. Theorem 35. Rotation index theorem (Hopf) The rotation index of a simple curve is ± Curves in Euclidean Space (3 Dimension). Let c : I R 3 be a regular curve. We want to define at each point c(t) an adaptive frame (orthonormal basis of vectors). We assume that ċ(t) and c(t) are linearly independent.

31 GEOMETRY AND TOPOLOGY 31 Definition Tangent unit vector, Normal unit vector, Binormal unit vector, Oscillating plane The tangent unit vector is given by The normal unit vector is given by e 1 (t) = ċ(t) ċ(t). e 2 (t) = c(t) c(t), e 1(t) e 1 (t) c(t) c(t), e 1 (t) e 1 (t) The binormal unit vector is given by e 3 (t) = e 1 (t) e 2 (t) The oscillating plane is given by span(e 1 (t), e 2 (t)) = span(ċ(t), c(t)). Lemma 36. The following hold (1) ė i (t), e j (t) + e i (t), ė j (t) = 0 (2) ė i (t), e i (t) = 0 = ė i (t) e i (t) (3) ė 1 (t), e 3 (t) = 0. Definition Curvature and Torsion Let c be a curve in R 3 with ċ(t) and c(t) linearly independent. Then the curvature of c is given by K(t) := ė 1 (t), e 2 (t) and the torsion of c is given by τ(t) := ė 2 (t), e 3 (t). We write K c and τ c to specify the curve c. Proposition 25. Frenet-Serret Formulas (1) ė 1 (t) = K(t)e 2 (t) (2) ė 2 (t) = K(t)e 1 (t) + τ(t)e 3 (t) (3) ė 3 (t) = τ(t)e 2 (t)

32 32 N. MOSHAYEDI Proposition 26. K c and τ c are independent of the parameterization of the oriented curve c. They can be expressed as follows: (1) (2) K c (t) = τ c (t) = ċ(t) c(t) ċ(t) 3... det(ċ(t), c(t), c (t)) ċ(t) c(t) 2 Definition Fundamental theorem of Curves Let c be a regular curve c : I R 3 such that ċ(t) and c(t) are linearly independent for all t I. Let T be an isometry of R 3. Then K c (t) = K T c (t) τ c (t) = τ T c (t). Conversely, let K and τ be smooth functions from I to R, with K strictly positive. Then there is a curve c parameterized by arc-length such that K(t) = K c (t) τ(t) = τ c (t). Any two such curves are selected by an isometry of R 3. Theorem 37. Total Curvature of space Curves Let c : R R 3 be a periodic space curve (regular parameterized by arc-length), with period ω. Then ω 0 K c (t)dt 2π. Equality can hold only if c P R 3 for some plane P R 3. Lemma 38. Let v S R 3. Then there exists t v [0, ω] such that ċ(t v ), v = Smooth Surfaces in R 3 We study here surfaces pictured as subspaces of R 3.

33 GEOMETRY AND TOPOLOGY 33 Definition Smooth Surface in R 3 A smooth surface in R 3 is a subset X R 3 such that for each point of x, there is an open neighborhood U X, an open set V R 2 and a map r : V R 3 such that r : V U is a homeomorphism. r is a local parameterization of X r is C. If r(u, v) = (x(u, v), y(u, v), z(u, v)), u, v R, x, y, z : R 2 R, then x, y, z have derivatives of all order in u, v. at each point of X, r u := r and r u v := r are linearly independent. v Remark Recall the definition of a surface we had in the topology part: (1) X is Hausdorff (2) X has a countable basis of its topology (3) X is locally homeomorphic to R 2, i.e. for each point of X, there is an open neighborhood U homeomorphic to an open subset V of R 2. That means a smooth surface in R 3 is a (topological) closed surface in the sense of the above. We can also make sense of the notion of a smooth surface without referring to R 3. Write ϕ U : U V for the homeomorphism in the definition of a topological surface. We call ϕ U a coordinate system on U. Then for ϕ U : U V and ϕ U : U V, we have the homeomorphisms ϕṽ Ṽ := ϕ U ϕ 1 U : Ṽ Ṽ, Ṽ = ϕ U (U U ) R 2, Ṽ = ϕ U (U U ) R 2. Definition Diffeomorphism A homeomorphism between subsets of R n that is C and such that its inverse is C is called a diffeomorphism. Definition Smooth Surface (general) A smooth surface is a closed topological surface such that the homeomorphisms ϕṽ Ṽ are all diffeomorphisms.

34 34 N. MOSHAYEDI Definition Smooth Map A smooth map between smooth surfaces X and Y is a continuous map f : X Y such that for each coordinate system ϕ U : U V with U X, x U and ϕ U : U V with U X, f(x) U, the composition is a smooth map. ϕ U f ϕ 1 U Example We have the following examples of smooth surfaces in R 3. Let e 1, e 2 and e 3 be an orthonormal basis of R 3. (1) A sphere of radius a: X = { p R 3 p = a}. We have the following parameterization: Let V 1 = ( π, π) (0, π), and Let U 1 = X \ { p R 3 c 1 e 2 + c 2 e 3, c 1 0} r 1 (u, v) = a sin u sin v e 2 + a cos u sin v e 2 + a cos v e 3. V 2 = ( π, π) (0, π), and U 2 = X \ { p R 3 c 1 e 1 + c 2 e 2, c 2 0} r 2 (u, v) = a cos v e 1 a cos u sin v e 2 + a sin u sin v e 3. It is not possible to cover the whole sphere with a simple parameterization r : V U. (2) A torus with outer radius a + b and inner radius a b. X = { p R 3 p = (a + b cos u)(cos v e 1 + sin v e 2 ) + b sin v e 3, (u, v) R 2 }. We have the following parameterization: Let V 1 = ( π, π) ( π, π),

35 GEOMETRY AND TOPOLOGY 35 U 1 = X \ { p R 3 p = (a b)(cos v e 1 + sin v e 2 ) or p = (a + b cos v) e 1 + b sin v e 3, v R} and r 1 (u, v) = (a + b cos u)(cos v e 1 + sin v e 2 ) + b sin u e 3. (3) A plane (con-compact). We have the following parameterization: r(u, v) = a + u b + v c, for all a, b, c R 3 with b and c linearly independent. Note that the plane is diffeomorphic to R 2, hence admits a global parameterization in terms of a single map r. Definition Change of Parameterization Let r : V R 3 be a local parameterization of a smooth surface X R 3. A change of parameterization is a diffeomorphism f : V V for some V R 2. The new parameterization is r = r f : V R 3. If f(x, y) = (u(x, y), v(x, y)), then ( r f) x = r u u x + r v v x ( r f) y = r u u y + r v v y i.e. ( ) ( ) ( ) ( ) ( r f)x ux v = x ru ru = J. ( r f) y u y v y r v r v As f 1 is differentiable, the Jacobian matrix J is invertible and {( r f) x, ( r f) y } are linearly independent iff { r u, r v } are linearly independent. Example The (x, y)-plane in R 3 admits the parameterization Define r(x, y) = x e 1 + y e 2. f : R ( π, π) R 2 \ R {0}, f(r, θ) = (r cos θ, r sin θ). The change of parameterization gives us another (local) parameterization of the (x, y)- plane: r f(r, θ) = r cos θ e 1 + r sin θ e 2.

36 36 N. MOSHAYEDI Definition Tangent Space The tangent space (or tangent plane) T p X of a surface X at a point p X is the vectorspace generated by { r u ( r 1 (p)), r v ( r 1 (p))}. Proposition 27. The tangent space is independent of the local parameterization r. Definition Unit Normals to the Surface The vectors ± n = ± ru rv r u r v are two unit normals to the surface. The two unit normals are orthogonal to the tangent plane. They are invariant under changes of parameterizations f that preserve the orientation of R The First Fundamental Form Let c : I R 3 be a regular curve in R 3 Let X be a surface, r : V U be a local parameterization. Assume that c(i) U. Then γ = ( r) 1 c is a curve in V. We have ċ = r v u + r v v where γ(t) = (u(t), v(t)). As { r u, r v } are linearly independent, the fact that c is regular implies that ( u, v) = γ 0 and hence γ is regular. Let us try to compute the length of c in the parameter space V, i.e. using γ instead of c. Let I = (a, b). Then (3) (4) (5) b a ċ(t) dt = = = b a b a b a ( ċ ċ)dt where E = r u r v, F = r u r v and G = r v r v. ( ru u + r v v) ( r u u + r v v)dt E u2 + 2F u v + G v 2 dt, Definition First Fundamental Form The first fundamental form of a surface X in R 3, relative to a local parameterization r : V U X, is the expression I r = E(du) 2 + 2F dudv + G(dv) 2, where E = r u r v, F = r u r v and G = r v r v. The first fundamental form is simply the quadratic form given by the standard scalar product on R 3, (Q v = v v) restrictted to the tangent space of X and expressed in the basis { r u, r v }. If v T p X with v = v 1 r u + v 2 r v, we should think of du and dv as linear forms on T p X selecting the component of v along r u and r v