Math 205C - Topology Midterm

Size: px
Start display at page:

Download "Math 205C - Topology Midterm"

Transcription

1 Math 205C - Topology Midterm Erin Pearse 1. a) State the definition of an n-dimensional topological (differentiable) manifold. An n-dimensional topological manifold is a topological space that is Hausdorff, has a countable basis at every point, and is locally Euclidean. That is, every point has a neighbourhood which is homeomorphic to an open set of R n. An n-dimensional differentiable manifold M is an n-dimensional topological manifold with a differentiable structure. That is, there is a collection of coordinate charts = {(U i, ϕ i )} which cover M such that i) x M, U x s.t. x U x and U x = V R n ii) For any two charts (U, ϕ) and (V, ψ), U V = ψ ϕ 1 and ϕ ψ 1 are diffeomorphisms of ϕ (U V ) and ψ (U V ) in R n. iii) U is maximal in the sense that any chart (U, ϕ) which compatible with in the sense of (ii) is included in. b) Describe all 1-dimensional manifolds. If a 1-dimensional manifold is compact, it is homeomorphic to S 1. If a 1- dimensional manifold is not compact, it is homeomorphic to R 1. Anything not falling into either category can readily be shown to be (i) not 1-dimensional, or (ii) not a topological manifold. c) Describe all closed orientable and non-orientable 2-dimensional manifolds. If a 2-dimensional closed manifold is orientable, then it is a sphere, a torus, or a connected sum of tori. That is, it is an n-genus torus (with a sphere corresponding to genus 0). If a 2-dimensional closed manifold is non-orientable, then it is a Klein bottle, projective plane, or connected sum of them. d) What are the fundamental groups of the manifolds in (b) and (c)? π 1 (S 1 ) = Z and S 1 has universal covering space R 1. π 1 (R 1 ) = {1} and R 1 is its own universal covering space. π 1 (S 2 ) = {1} and S 2 has universal covering space R 2. π 1 (T 2 ) = Z Z and T 2 has universal covering space R 2. π 1 (# g i=1 T 2 ) = Z 2g and # g i=1 T 2 has universal covering space R 2. π 1 (K) = F (a,b) and K has universal covering space {aba 1 b} R2. ( π ) 1 RP 2 = Z2 and RP 2 has universal covering space S 2.

2 2 Math 205C - Topology Midterm Erin Pearse 2. Explain why each of the following figures either is or is not a topological manifold. a) This object is clearly 1-dimensional. Take x to be an intersection point with open nbd U as shown. Assuming this space has the natural (subspace) topology, U would have to be homeomorphic to an open set V R 1. Since U {x} consists of three connected components, and for V {f(x)} consists of only two, f : U V cannot be a homeomorphism. This is explained in a little better detail in (v), below. b) By the same argument as above, any open nbd of the intersection point x cannot be homemorphic to an open set of R 1. c) Though not a differentiable manifold, this space is clearly homeomorphic to (a, b) R 1 and is thus a 1-dimensional topological manifold. d) This space is not Hausdorff, as any open set containing x also contains y. Since it is not Hausdorff, it fails to be a topological manifold. e) Any point of this space has a nbd homeomorphic to an open subset V R 2, unless it lies in the intersection of the two planes. In this case, an open nbd U of x looks like two intersecting plates. Let l be the line that forms the intersection, and suppose f : U V R 2 is a homeomorphism. But U\l consists of 4 distinct connected components, and f(u\l) can consist of at most two connected components. To see this, note that for f(l) to separate R 2 into more components, f l would not be injective. Thus, this space is not a topological manifold.

3 Math 205C - Topology Midterm Erin Pearse 3 3. a) Prove that R 1 cannot be homeomorphic to R 2, for n 2. Let f : R 1 R n, and let U = R 1 {x} where x R 1. Then if f were a homeomorphism, f R {x} : R {x} R n would also be a homeomorphism. Then, R {x} is disconnected f (R {x}) is disconnected. But for n 2, R n will always be connected! < Therefore, such a homeomorphism cannot exist. b) Try to prove that R n cannot be homeomorphic to R m, for n m. Begin by taking n < m. Then we imbed f : R n R m by inclusion. Note that any open set of R m must have dimension m. f(r n ) = R n has dimension n, so clearly R n cannot be an open subset of R m. Thus, f cannot be a homeomorphism, because it takes an open set to a non-open set a) Give an example of a closed 3-dimensional manifold whose fundamental group is (Z, +). i) S 2 S 1 b) Give two examples of a closed simply-connected 4-dimensional manifold. i) S 2 S 2 ii) S 4 c) Give two examples of a closed, simply-connected 3-dimensional manifold. i) S 3 ii) Umm... 1 I understand why this argument is wrong - the result is due to a confusion of the topologies on R n relative to the homeomorphism and to R m. However, this problem asks us to try to prove the theorem, and this was the best I could come up with, short of re-deriving Brouwer s Invariance of Domain from scratch.

4 4 Math 205C - Topology Midterm Erin Pearse 5. Let M be a topological (differentiable) manifold with an infinite fundamental group. Prove that its universal covering M is a non-compact topological (differentiable) manifold. Since M is a universal covering, it is path-connected, so for any point m 0 M, we can use the lifting correspondence to establish a bijection π 1 (M) Φ : ( ) p 1 (m 0 ) π 1 M (cf. [Munk] Thm. 54.6b). Then we immediately have that p 1 (m 0 ) π 1 (M) = ( ) by bijection π 1 M = π 1 (M) {1} = π 1 (M) M is simply connected = by hypothesis Now let U be an open nbd of m 0 so that m 0 U M. Because p is a covering map, we have p 1 (U) = i I V i where I is an infinite indexing set and p Vi : V i U is a homeomorphism. Let {x i } i=1 by any countable sequence of points in p 1 (m 0 ). Then {x i } is an infinite sequence in M, and x i V i i. Since the definition of covering map guarantees that all the V i are disjoint, {x i } clearly cannot have any limit points. Thus, by the equivalence of compactness and limit point compactness in metrizable spaces, M must not be compact.

5 Math 205C - Topology Midterm Erin Pearse 5 6. Define the following terms: a) rank of a differentiable mapping The rank of a differentiable mapping F at p is the rank at a = ϕ(p) of the Jacobian matrix 1 x 1 1 x n f a =..... m x 1 b) immersion An immersion is a differentiable mapping f : M N such that dim M = rank f at every point of M. In this case, f : T p (M) T ϕ(p) (M) is injective. c) submersion An submersion is a differentiable mapping f : M N such that dim N = rank f at every point of N. In this case, f : T p (M) T ϕ(p) (M) is surjective. d) imbedding An imbedding is an injective immersion f : M N which is homeomorphic to its image f(m) N. In this case, the topology on f(m) induced by f and the subspace topology on f(m) induced by N are identical. e) submanifold (i.e., immersed submanifold) A submanifold or immersed submanifold is just the image of an immersion, that is, the image of a manifold M under an immersion. In this case, the topology on f(m) is taken as being induced by the topology on M, via the immersive map. f) imbedded submanifold An imbedded submanifold is the image of an imbedding, that is, the image of a manifold under an immersion. m x n a

6 6 Math 205C - Topology Midterm Erin Pearse 7. a) Is an injective immersion an imbedding? If not, give two examples. i) Define the figure-eight map G : R R 2 as G (t) = ( 2 cos ( π + 2 arctan t), sin 2 ( π + 2 arctan t)) 2 2 G 1 (t) will not be continuous at (0, 0) for reasons similar to those discussed in #2(b,e). The essential ideal is that any sufficiently small neighbourhood of (0, 0) which is open in the subspace topology of R 2, will be mapped by G 1 (t) to a disjoint union of three subsets of R. ii) Define a variant of the topologist s sine curve as ( 1, sin πt) t > 1 t 2 F (t) = (smooth) 0 t 1 2 (0, t + 3) t < 0 where smooth is a smooth arc connecting the other two sections of the graph such that the entire set is C. (see [Boot] III.4.10, Fig. III.8) The idea of this example is similar to the previous one. F 1 (t) will not be continuous at any point in the set {(0, y) R 2. 1 < y < 1}. b) Let f : M N be a injective immersion. Is f(m) a closed subset in N? If not, give an example. No, f(m) need not be closed in N. Define f : R R 2 by f (x) = ( 2 arctan x, 0) π so that f (R) = {(x, 0). 1 < x < 1} Since the limit points ( 1, 0) and (1, 0) are not contained in f (R), f (R) must not be closed. c) Let f : M N be an imbedding. Is f(m) a closed subset in N? If not, give an example. No, f(m) need not be closed in N. Define f : R 2 R 2 by f (x, y) = ( 2 arctan x, 2 arctan y) π π so that f ( R 2) = {(x, y) R 2. 1 < x, y < 1} Then f (R 2 ) is an open rectangle in R 2 and clearly not closed.

7 Math 205C - Topology Midterm Erin Pearse 7 d) Let f : M N be an injective immersion, and consider f(m) as endowed with the subspace topology from N. i) Is f : M f (M) a continuous map? An immersion is by definition differentiable, and continuity is a prerequisite for differentiability. So f is clearly continuous onto its image. ii) Is f 1 : f (M) M a continuous map? No, not in general. The injective immersion f : R F of R as a figureeight F R 2 is a counterexample. Let the intersection point of F be at the origin, and let B ε (0) be the ball of radius ε centered at the origin. For small enough ε, it is clear that any point x on two of the four branches of B ε (0) F, f 1 will take x outside of ( 1, 1). In other words, f 1 is not continuous at 0 R 2. iii) What happens when M is a compact manifold? When M is compact, we can show that f must be proper, from which it follows (by #8.b) that f is an imbedding, and that f(m) is a regular submanifold. Let K M be a closed subset of M. Then we have K closed K compact (M is compact) f(k) compact (f is continuous) f(k) closed (N is Hausdorff) This shows that f is a closed map. Now let C be a compact subset of N, and we will show f 1 (C) is compact. Let {U α } α A be an open covering of f 1 (C). For any y C, α such that f 1 (y) U α. Denote by U y the U α that contains f 1 (y). Now define V y = N f (M U y ) and note that V y is an open set containing y. V y is an open because f (and hence f (M U y )) is closed. Now C is compact, and {V y } covers C, so we can find a finite subcover {V yi } n i=1. Then f 1 (C) f 1 (V y1 )... f 1 (V yn ) because U y1... U yn f 1 (V y ) = f 1 (Y f (X U y )) f 1 (Y ) (X U y ) = X (X U y ) = U y f 1 (V y ) U y. But then {U yi } n i=1 is a finite subcover of f 1 (C), so f 1 (C) is compact. Hence, f is proper. (see also [Bred] I.7.13 and [Boot] III.5.7)

8 8 Math 205C - Topology Midterm Erin Pearse 8. a) Let f : M N be a proper continuous map between two topological manifolds M and N. Prove that f(m) is a closed subset of N. Since M,N are manifolds, they are locally compact. Let Ṁ denote the 1-point compactification of M. Note that if M is already compact, then the new point m is an isolated point of M that is clopen, i.e., both closed and open. 2 Similarly, take (Ṅ = N n) to be the 1-point compactification of M. Define f ( m ) = n and note that this extends f : Ṁ Ṅ continuously as follows: For U open in N, f 1 (U) is open in M by continuity. For U = Ṁ K, we have f 1 (U) = f 1 1 (Ṅ K) = f (Ṅ) f 1 (K) = Ṁ f 1 (K), which is open in Ṁ by proper. Now we have that Ṁ is compact by construction, so f(ṁ) is compact as the continuous image of a compact set. Since N, and hence Ṅ, is Hausdorff, this gives us that f(ṁ) is closed in Ṅ as a compact subset of a Hausdorff space. But N is closed in Ṅ, so then f(ṁ) N is closed in N by subspace topology. But f(ṁ) = f (M m) = f (M) f ( m ) = f (M) n. Thus, f(ṁ) N = (f (M) n) N = f (M). Hence f(m) is closed in N. Note: for D closed in M, D Ṁ implies that f(d) is compact, and hence closed in Ṅ. So this proof actually shows that f is a closed map! b) Let f : M N be a proper injective immersion of differentiable manifolds M and N. Prove that f is an imbedding and f(m) is a regular submanifold and a closed subset of M, and conversely. At the end of part (a), it was shown that f is a closed map. f is given to be injective, so f is a bijection when restricted to its image as f : M f (M). Then f : M f (M) is also an open map, which is equivalent to saying f 1 is continuous. Then f a bijection and f, f 1 continuous shows that f : M f (M) is a homeomorphism and hence that f : M N is an imbedding. Note that f(m) is closed in N by part (a), and that ([Boot] III.5.5, p.78) implies that f(m) is regular. f is given to be an imedding, so we immediately have that f is an injective immersion, by definition. To see that f is proper, choose K N compact. Then N Hausdorff implies K is closed. Since f(m) is closed by (a), this shows K f (M) is a closed subset of K, and hence is also compact. Now f is an imbedding, so f 1 is continuous. Thus, f 1 (K) is a continuous image of a compact set, and is hence compact. So we have shown K compact f 1 (K) compact. 2 This follows from the definition of the topology of Ṁ: U is open in Ṁ iff U is open in M or U = Ṁ K for some compact set K M.

9 Math 205C - Topology Midterm Erin Pearse 9 9. Let f : R 3 R 4 be a mapping given by f (x 1, x 2, x 3 ) = (x 1 x 2, x 1 x 3, x 2 x 3, x 1 + x 2 + x 3 ). a) Compute the Jacobian of f at (1, 1, 1) x 1 x 2 x 3 x 2 x 1 0 x = x 1 x 2 x 3 x 3 0 x = x 1 x 2 x 3 0 x 3 x x = (1,1,1) = x 1 x 2 x = = = b) What is the rank of f at (1, 1, 1)? ( By the above calculation, rank x ) (1,1,1) = 3. c) Could you find a point of R 3 such that the rank of f is less than 3? On the x 1 -axis, when x 2 and x 3 are 0, we have: 0 a 0 x = 0 0 a (a,0,0) 0 0 0, which still has rank 3 for any a. Hence, we will only have rank less than three at the origin: x = (0,0,0) 0 0 0, which has rank 1. d) What are the points in R 3 at which f is immersive? f is immersive on R 3 \{0}.

10 10 Math 205C - Topology Midterm Erin Pearse 10. State and understand the Inverse Function Theorem, the Implicit Function Theorem, and the Rank Theorem. a) The Inverse Function Theorem. Let W be an open subset of R n and f : W R n a C r mapping for r Z +. If a W and f (a) is nonsingular, then there exists an open neighbourhood U of a in W such that V = f(u) is open and f : U V is a C r diffeomorphism. If x U and y = f(x), then the derivatives of f 1 at y are given by ( f 1 ) (y) = (f (x)) 1 where (f (x)) 1 is the inverse matrix of f (x). b) The Implicit Function Theorem. (Surjective form). Let U R m be an open set and f : U R n a C r map, r Z +. Let p U, f(p) = 0, and suppose that f x is surjective. Then there exists a local diffeomorphism ϕ of R m at 0 such that ϕ(0) = p and f ϕ (x 1,..., x m ) = (x 1,..., x n ). (Injective form). Let U R m be an open set and f : U R n a C r map, r Z +. Let q R n be such that 0 f 1 (q), and suppose that f 0 injective. Then there exists a local diffeomorphism ψ of R n at q such that ψ(q) = 0 and ψ f(x) = (x 1,..., x m, 0,..., 0). c) The Rank Theorem. Let A 0 R n, B 0 R m be open sets, f : A 0 B 0 be a C r mapping, and suppose the rank of f on A 0 to be equal to k. If a A 0 and b = f(a), then there exist open sets A A 0 and B B 0 with a A and b B. Also, there exist open sets U R n and V R m and C r diffeomorphisms g : A U, h : B V such that h f g 1 (U) V and such that h f g 1 (x 1, x 2,..., x n ) = (x 1, x 2,..., x n, 0,..., 0).

11 Math 205C - Topology Midterm Erin Pearse Let M and N be two n-dimensional differentiable manifolds. a) Suppose f : M N is an immersion. Prove that f is an open map. Let W M be open and pick y f(w ). Then y = f(x) for some x W. Since f is an immersion, we know Df(x) is nonsingular. By the Inverse Function Theorem, x has an open neighbourhood U W such that f (U) is open. Then U W implies f(u) f(w ) is an open neighbourhood of y. Since we can do this for any y, f(w ) is open. Alternative proof: Let W M be open and consider f(w ). Pick y f(w ) so y = f(x) for some x W. Then x has a neighbourhood U such that f U is an imbedding of U in N, by ([Boot] III.4.12, p.74). Thus, f(u) f(w ) is an open neighbourhood of y. Since we can do this for any y, f(w ) is open. b) Suppose f : M N is an immersion and M is compact. Prove that f is a surjection. M is open in M trivially, so f(m) is open in N by (a). Then M is compact by hypothesis, so we know f(m) is compact and hence closed, because N is Hausdorff. Evidently f(m) is both open and closed. Observe that N is connected. 3 Since the only clopen subsets of a connected space are the entire space and the empty set, it must be the case that f(m) = N. 3 At this point, we use the fact that we are considering all manifolds to be connected. If this condition is not included in the definition, this theorem can readily be demonstrated to be false. For example, let M be S 1 and let N be a disjoint union of two copies of S 1. Then any immersion of M in N will not be surjective.

12 12 Math 205C - Topology Midterm Erin Pearse 12. a) What is a C partition of unity? A C partition of unity on M is a collection of C functions {f α } defined on M and satisfying: i) f α 0 on M ii) {supp (f α )} forms a locally finite covering of M iii) α f α (x) = 1, x M b) Let p q be two distinct points on a differentiable manifold M. Prove that there exists a C function f : M R such that f(p) = 100 and f(q) = Let g be the imbedding of M into R n, as guaranteed by the Whitney Imbedding Theorem. Then let b c, ε (x) : R n R be the bump function centered at c with support {x R n. x c ε}, as defined in class. Then set ε = 1 g(p) g(q) 2 and define b as follows: b (x) = 100b p,ε (x) b q,ε (x) Finally, we define f = b g. Because C (R n ) is an algebra, f is clearly C. Alternative proof: As a manifold, M is Hausdorff, so we can find two disjoint open sets U and V such that p U and q V. Then since M is locally compact, we can find a neighbourhood C of x such that C is compact and C U, and we can find a neighbourhood D of y such that D is compact and D V. By ([Boot] III.3.4, P.67) there exist C functions 0, x M\U 0, x M\V b C (x) = 0 < α < 1, x U\C and b D (x) = 0 < β < 1, x V \D. 1, x C 1, x D Now define f (x) = 100b C (x) b D (x).

13 Math 205C - Topology Midterm Erin Pearse a) State the Whitney Imbedding Theorem The easy Whitney Imbedding Theorem states: Any differentiable manifold M of dimension n may be imbedded differentiably as a closed submanifold of R 2n+1. The hard Whitney Imbedding Theorem states: For n > 0, every paracompact Hausdorff n-manifold can be imbedded into R 2n. Furthermore, it may be immersed in R 2n 1 if n > 1. b) Let M be a compact differentiable manifold of dimension n. Prove that there exists an imbedding f : M R n for sufficiently large n, without using (a). Take n = dim M and let {A α } be any open covering of M. By ([Boot] V.4.1), there exists a countable, locally finite refinement {U i } i=1 consisting of coordinate neighbourhoods (U i, ϕ i ) with i) ϕ i (U i ) = B n 3 (0) i, and ii) for V i = ϕ 1 i (B1 n (0)) U i, M = i=1 V i Since M is compact, we need consider only a finite set of coordinate charts {(U i, V i, ϕ i )} k i=1. Now by ([Boot] V.4.4), we can take a subordinate C partition of unity {g i } k i=1 such that g i = 1 on V i. Define the C maps f i : M R n by { g i ϕ i, x U i f i (x) = 0, otherwise so that each f i is immersive on V i. Now for R N = R n+1 R n+1 = R (n+1)k and F i = (f i, g i ), define F : M R N by F (p) = (F 1 (p),..., F k (p)) F is componentwise C and thus C. F is injective. Pick x y, with y V i. case i) x V i. Then f i Vi = ϕ i Vi = F (x) F (y) (ϕ i is injective) case ii) x / V i. Then g i (y) = 1 g i (x) = F (x) F (y) F is an immersion. For x M, x V i for some i. Then = g i, and hence g, is immersive at x, as in the proof of #16. Now we can apply #7(d)iii to obtain the desired result. (See [Boot], p.196)

14 14 Math 205C - Topology Midterm Erin Pearse 14. Let M be a differentiable manifold. Prove that there exists a proper differentiable map f : M R n, for any positive integer n. Pick some point x 0 M and define f(x) = d(x 0, x), where d is the Riemannian metric. This is justified, because it is possible to define a C Riemannian metric on every C manifold (cf. [Boot] V.4.5, p.195), and because a Riemannian manifold (that is, manifold on which a Riemannian metric Φ is defined) is a complete metric space in which the metric topology and manifold topology coincide (cf. [Boot] V.3.1, p.189). f is given explicitly by f (x) = inf c(t) D 1 { v u ( Φ ( dp dt, dp dt )) 1/2 dt. c (u) = x 0, c (v) = x, u t v} Then f : M R + is a metric and hence differentiable. For any compact interval [a, b] R +, its preimage will be an annulus about x 0 (i.e., f 1 ([a, b]) = {x R. a x x 0 b} ). This is enough to show that the preimage of any compact set is compact. Since f : M R is proper and differentiable, we can define the imbedding g : R R n by inclusion. Then g f will work, for any n. Alternative proof: To avoid the use of Riemannian metrics, this proof can be adapted as follows: use the Whitney embedding theorem to embed M in some R N, then replace Φ with the standard n-dimensional Euclidean metric. The rest of the proof remains essentially the same.

15 Math 205C - Topology Midterm Erin Pearse Prove that there exists no immersion f : S n R n. S n is compact and f is continuous, so f(s n ) is compact. Since R n is Hausdorff, f(s n ) is also closed in R n. Following ([Munk] p.102), we define the boundary of a set A X by A = A (X A) Then we can pick a point x (S n ) and then also a point y f 1 (x) f 1 ( (S n )). Now, if f really were an immersion, the Df would be nonsingular everywhere. In particular, Df would be nonsingular at x. Then we can find an open nbd U S n of x such that V = f(u) is open in R n by the Inverse Function Theorem. Yet f(u) is clearly not open, because x f(u) and any open nbd of x is not contained in f(u) by the choice x (S n ). To see this, note that the definition of boundary makes x a limit point of (R n f (S n )). Alternative proof: Note that S n and R n are manifolds of the same dimension, and S n is compact. Then we know by #11 that any immersion f : S n R n would be a surjection and that f is an open map. Let {U α } a A be an open covering of R n. Then {f 1 (U α )} a A is an open covering of S n, so it must have a finite subcover {f 1 (U αi )} n i=1. Thus, {f (f 1 (U αi ))} n i=1 = {U α i } n i=1 (equality because f is surjective) shows that any open covering of R n has a finite subcovering. < (R n is not compact)

16 16 Math 205C - Topology Midterm Erin Pearse 16. Let f : M R n be an injective immersion. Prove that there exists an imbedding F : M R n+1 such that f(m) is a closed subset of R n+1. We have f (x) = (f 1 (x), f 2 (x),..., f n (x)), and we know from the proof of #14 that we can find a proper differentiable map g : M R. Define F (x) = (f 1 (x), f 2 (x),..., f n (x), g (x)) So that F : M R n+1. We will show that F is a proper injective immersion. Then the result will follow by #8(b). F is proper. Let {x i } i=1 be any divergent sequence in M. Then {g (x i)} i=1 will be a divergent sequence because g is proper. Then it must also be the case that {F (x i )} i=1 = {(f 1 (x i ),..., f n (x i ), g (x i ))} i=1 is a divergent sequence. This shows that F is proper. F is injective. Let m 1 m 2 be distinct points of M. Then f is injective, so f (m 1 ) f (m 2 ). This implies F (m 1 ) = (f (m 1 ), g (m 1 )) (f (m 2 ), g (m 2 )) = F (m 2 ). F is an immersion. First we note that f : M N is an immersion Df : T p (M) T f(p) (N) is injective. ( ) Pick v T p (M) such that DF (v) = 0. Then 1 x 1 1 x m DF (v) =... v 1.. n x 1 n. = x m g g v m x 1 x m This can only be true if 1 v dx Df (v) = n dx. v = 1 v dx. n v dx g v dx = And since f is an immersion, we know by ( ) that Df is injective. Hence, this can only happen when v = [0,..., 0]. This shows that 0. 0 DF (v) = 0 v = 0, i.e., that DF is injective. Then by ( ) again, F is an immersion

17 Math 205C - Topology Midterm Erin Pearse 17 Proof of ( ) f : M N is an immersion Df : T p (M) T f(p) (N) is injective. Note that dim M = dim T p (M) and dim V = rank A + dim (ker A), for any linear mapping A defined on a vector space V. Since Df : T p (M) T f(p) (N) is linear, this gives Now we have dim T p (M) = rank Df + dim (ker Df) f : M N is an immersion dim M = rank Df dim (ker Df) = 0 ker Df = {0} Df is injective.

18 18 Math 205C - Topology Midterm Erin Pearse References [Bred] [Boot] [Hirs] [Mass] [Miln] [Munk] Bredon, Glen E. (1993) Topology and Geometry. Springer-Verlag. Boothby, William M. (1986) An Introduction to Differentiable Manifolds and Riemannian Geometry (Second Edition). Academic Press. Hirsch, Morris W. (1976) Differential Topology. Springer-Verlag. Massey, William S. (1967) Algebraic Topology: An Introduction. Springer-Verlag. Milnor, John W. (1965) Topology from the Differentiable Viewpoint. Princeton University Press. Munkres, James R. (2000) Topology (Second Edition). Prentice Hall.

Math General Topology Fall 2012 Homework 8 Solutions

Math General Topology Fall 2012 Homework 8 Solutions Math 535 - General Topology Fall 2012 Homework 8 Solutions Problem 1. (Willard Exercise 19B.1) Show that the one-point compactification of R n is homeomorphic to the n-dimensional sphere S n. Note that

More information

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X. Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

More information

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov

More information

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

More information

Topological properties

Topological properties CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

More information

Math 215B: Solutions 3

Math 215B: Solutions 3 Math 215B: Solutions 3 (1) For this problem you may assume the classification of smooth one-dimensional manifolds: Any compact smooth one-dimensional manifold is diffeomorphic to a finite disjoint union

More information

Math 637 Topology Paulo Lima-Filho. Problem List I. b. Show that a contractible space is path connected.

Math 637 Topology Paulo Lima-Filho. Problem List I. b. Show that a contractible space is path connected. Problem List I Problem 1. A space X is said to be contractible if the identiy map i X : X X is nullhomotopic. a. Show that any convex subset of R n is contractible. b. Show that a contractible space is

More information

Math 396. Bijectivity vs. isomorphism

Math 396. Bijectivity vs. isomorphism Math 396. Bijectivity vs. isomorphism 1. Motivation Let f : X Y be a C p map between two C p -premanifolds with corners, with 1 p. Assuming f is bijective, we would like a criterion to tell us that f 1

More information

Geometry 2: Manifolds and sheaves

Geometry 2: Manifolds and sheaves Rules:Exam problems would be similar to ones marked with! sign. It is recommended to solve all unmarked and!-problems or to find the solution online. It s better to do it in order starting from the beginning,

More information

Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds

Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds MA 755 Fall 05. Notes #1. I. Kogan. Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds Definition 1 An n-dimensional C k -differentiable manifold

More information

Chapter 1. Smooth Manifolds

Chapter 1. Smooth Manifolds Chapter 1. Smooth Manifolds Theorem 1. [Exercise 1.18] Let M be a topological manifold. Then any two smooth atlases for M determine the same smooth structure if and only if their union is a smooth atlas.

More information

1 k x k. d(x, y) =sup k. y k = max

1 k x k. d(x, y) =sup k. y k = max 1 Lecture 13: October 8 Urysohn s metrization theorem. Today, I want to explain some applications of Urysohn s lemma. The first one has to do with the problem of characterizing metric spaces among all

More information

4 Countability axioms

4 Countability axioms 4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said

More information

Theorem 3.11 (Equidimensional Sard). Let f : M N be a C 1 map of n-manifolds, and let C M be the set of critical points. Then f (C) has measure zero.

Theorem 3.11 (Equidimensional Sard). Let f : M N be a C 1 map of n-manifolds, and let C M be the set of critical points. Then f (C) has measure zero. Now we investigate the measure of the critical values of a map f : M N where dim M = dim N. Of course the set of critical points need not have measure zero, but we shall see that because the values of

More information

M4P52 Manifolds, 2016 Problem Sheet 1

M4P52 Manifolds, 2016 Problem Sheet 1 Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete

More information

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds

More information

Math 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim

Math 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each).

More information

Solve EACH of the exercises 1-3

Solve EACH of the exercises 1-3 Topology Ph.D. Entrance Exam, August 2011 Write a solution of each exercise on a separate page. Solve EACH of the exercises 1-3 Ex. 1. Let X and Y be Hausdorff topological spaces and let f: X Y be continuous.

More information

i = f iα : φ i (U i ) ψ α (V α ) which satisfy 1 ) Df iα = Df jβ D(φ j φ 1 i ). (39)

i = f iα : φ i (U i ) ψ α (V α ) which satisfy 1 ) Df iα = Df jβ D(φ j φ 1 i ). (39) 2.3 The derivative A description of the tangent bundle is not complete without defining the derivative of a general smooth map of manifolds f : M N. Such a map may be defined locally in charts (U i, φ

More information

1 The Local-to-Global Lemma

1 The Local-to-Global Lemma Point-Set Topology Connectedness: Lecture 2 1 The Local-to-Global Lemma In the world of advanced mathematics, we are often interested in comparing the local properties of a space to its global properties.

More information

fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))

fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f)) 1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical

More information

Math General Topology Fall 2012 Homework 13 Solutions

Math General Topology Fall 2012 Homework 13 Solutions Math 535 - General Topology Fall 2012 Homework 13 Solutions Note: In this problem set, function spaces are endowed with the compact-open topology unless otherwise noted. Problem 1. Let X be a compact topological

More information

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology

More information

LECTURE 9: THE WHITNEY EMBEDDING THEOREM

LECTURE 9: THE WHITNEY EMBEDDING THEOREM LECTURE 9: THE WHITNEY EMBEDDING THEOREM Historically, the word manifold (Mannigfaltigkeit in German) first appeared in Riemann s doctoral thesis in 1851. At the early times, manifolds are defined extrinsically:

More information

Notes on Spivak, Differential Geometry, vol 1.

Notes on Spivak, Differential Geometry, vol 1. Notes on Spivak, Differential Geometry, vol 1. Chapter 1. Chapter 1 deals with topological manifolds. There is some discussion about more subtle topological aspects (pp. 2 7) which we can gloss over. A

More information

General Topology. Summer Term Michael Kunzinger

General Topology. Summer Term Michael Kunzinger General Topology Summer Term 2016 Michael Kunzinger michael.kunzinger@univie.ac.at Universität Wien Fakultät für Mathematik Oskar-Morgenstern-Platz 1 A-1090 Wien Preface These are lecture notes for a

More information

Let X be a topological space. We want it to look locally like C. So we make the following definition.

Let X be a topological space. We want it to look locally like C. So we make the following definition. February 17, 2010 1 Riemann surfaces 1.1 Definitions and examples Let X be a topological space. We want it to look locally like C. So we make the following definition. Definition 1. A complex chart on

More information

Bredon, Introduction to compact transformation groups, Academic Press

Bredon, Introduction to compact transformation groups, Academic Press 1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions

More information

Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009.

Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009. Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009. Solutions (1) Let Γ be a discrete group acting on a manifold M. (a) Define what it means for Γ to act freely. Solution: Γ acts

More information

LECTURE 11: TRANSVERSALITY

LECTURE 11: TRANSVERSALITY LECTURE 11: TRANSVERSALITY Let f : M N be a smooth map. In the past three lectures, we are mainly studying the image of f, especially when f is an embedding. Today we would like to study the pre-image

More information

BROUWER FIXED POINT THEOREM. Contents 1. Introduction 1 2. Preliminaries 1 3. Brouwer fixed point theorem 3 Acknowledgments 8 References 8

BROUWER FIXED POINT THEOREM. Contents 1. Introduction 1 2. Preliminaries 1 3. Brouwer fixed point theorem 3 Acknowledgments 8 References 8 BROUWER FIXED POINT THEOREM DANIELE CARATELLI Abstract. This paper aims at proving the Brouwer fixed point theorem for smooth maps. The theorem states that any continuous (smooth in our proof) function

More information

Math 190: Fall 2014 Homework 4 Solutions Due 5:00pm on Friday 11/7/2014

Math 190: Fall 2014 Homework 4 Solutions Due 5:00pm on Friday 11/7/2014 Math 90: Fall 04 Homework 4 Solutions Due 5:00pm on Friday /7/04 Problem : Recall that S n denotes the n-dimensional unit sphere: S n = {(x 0, x,..., x n ) R n+ : x 0 + x + + x n = }. Let N S n denote

More information

Terse Notes on Riemannian Geometry

Terse Notes on Riemannian Geometry Terse Notes on Riemannian Geometry Tom Fletcher January 26, 2010 These notes cover the basics of Riemannian geometry, Lie groups, and symmetric spaces. This is just a listing of the basic definitions and

More information

Good Problems. Math 641

Good Problems. Math 641 Math 641 Good Problems Questions get two ratings: A number which is relevance to the course material, a measure of how much I expect you to be prepared to do such a problem on the exam. 3 means of course

More information

LECTURE 3: SMOOTH FUNCTIONS

LECTURE 3: SMOOTH FUNCTIONS LECTURE 3: SMOOTH FUNCTIONS Let M be a smooth manifold. 1. Smooth Functions Definition 1.1. We say a function f : M R is smooth if for any chart {ϕ α, U α, V α } in A that defines the smooth structure

More information

Maths 212: Homework Solutions

Maths 212: Homework Solutions Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

More information

1. Simplify the following. Solution: = {0} Hint: glossary: there is for all : such that & and

1. Simplify the following. Solution: = {0} Hint: glossary: there is for all : such that & and Topology MT434P Problems/Homework Recommended Reading: Munkres, J.R. Topology Hatcher, A. Algebraic Topology, http://www.math.cornell.edu/ hatcher/at/atpage.html For those who have a lot of outstanding

More information

Then A n W n, A n is compact and W n is open. (We take W 0 =(

Then A n W n, A n is compact and W n is open. (We take W 0 =( 9. Mon, Nov. 4 Another related concept is that of paracompactness. This is especially important in the theory of manifolds and vector bundles. We make a couple of preliminary definitions first. Definition

More information

LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES

LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES 1. Proper actions Suppose G acts on M smoothly, and m M. Then the orbit of G through m is G m = {g m g G}. If m, m lies in the same orbit, i.e. m = g m for

More information

Defn 3.1: An n-manifold, M, is a topological space with the following properties:

Defn 3.1: An n-manifold, M, is a topological space with the following properties: Chapter 1 all sections 1.3 Defn: M is locally Euclidean of dimension n if for all p M, there exists an open set U p such that p U p and there exists a homeomorphism f p : U p V p where V p R n. (U p, f)

More information

Topology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng

Topology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng Topology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng Remark 0.1. This is a solution Manuel to the topology questions of the Topology Geometry

More information

DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES. September 25, 2015

DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES. September 25, 2015 DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES MAGGIE MILLER September 25, 2015 1. 09/16/2015 1.1. Textbooks. Textbooks relevant to this class are Riemannian Geometry by do Carmo Riemannian Geometry

More information

Notes for Math 535 Differential Geometry Spring Francis Bonahon. Department of Mathematics, University of Southern California

Notes for Math 535 Differential Geometry Spring Francis Bonahon. Department of Mathematics, University of Southern California Notes for Math 535 Differential Geometry Spring 2016 Francis Bonahon Department of Mathematics, University of Southern California Date of this version: April 27, 2016 c Francis Bonahon 2016 CHAPTER 1 A

More information

Foliations of Three Dimensional Manifolds

Foliations of Three Dimensional Manifolds Foliations of Three Dimensional Manifolds M. H. Vartanian December 17, 2007 Abstract The theory of foliations began with a question by H. Hopf in the 1930 s: Does there exist on S 3 a completely integrable

More information

SOLUTIONS TO THE FINAL EXAM

SOLUTIONS TO THE FINAL EXAM SOLUTIONS TO THE FINAL EXAM Short questions 1 point each) Give a brief definition for each of the following six concepts: 1) normal for topological spaces) 2) path connected 3) homeomorphism 4) covering

More information

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1 MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and

More information

1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0

1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0 1. Classification of 1-manifolds Theorem 1.1. Let M be a connected 1 manifold. Then M is diffeomorphic either to [0, 1], [0, 1), (0, 1), or S 1. We know that none of these four manifolds are not diffeomorphic

More information

Math 6455 Oct 10, Differential Geometry I Fall 2006, Georgia Tech. Integration on Manifolds, Volume, and Partitions of Unity

Math 6455 Oct 10, Differential Geometry I Fall 2006, Georgia Tech. Integration on Manifolds, Volume, and Partitions of Unity Math 6455 Oct 10, 2006 1 Differential Geometry I Fall 2006, Georgia Tech Lecture Notes 13 Integration on Manifolds, Volume, and Partitions of Unity Suppose that we have an orientable Riemannian manifold

More information

SARD S THEOREM ALEX WRIGHT

SARD S THEOREM ALEX WRIGHT SARD S THEOREM ALEX WRIGHT Abstract. A proof of Sard s Theorem is presented, and applications to the Whitney Embedding and Immersion Theorems, the existence of Morse functions, and the General Position

More information

1 z. = 2 w =. As a result, 1 + u 2. 2u 2. 2u u 2 ) = u is smooth as well., ϕ 2 ([z 1 : z 2 ]) = z 1

1 z. = 2 w =. As a result, 1 + u 2. 2u 2. 2u u 2 ) = u is smooth as well., ϕ 2 ([z 1 : z 2 ]) = z 1 KOÇ UNIVERSITY FALL 011 MATH 554 MANIFOLDS MIDTERM 1 OCTOBER 7 INSTRUCTOR: BURAK OZBAGCI 180 Minutes Solutions by Fatih Çelik PROBLEM 1 (0 points): Let N = (0 0 1) be the north pole in the sphere S R 3

More information

Many of the exercises are taken from the books referred at the end of the document.

Many of the exercises are taken from the books referred at the end of the document. Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the

More information

Integration and Manifolds

Integration and Manifolds Integration and Manifolds Course No. 100 311 Fall 2007 Michael Stoll Contents 1. Manifolds 2 2. Differentiable Maps and Tangent Spaces 8 3. Vector Bundles and the Tangent Bundle 13 4. Orientation and Orientability

More information

NOTES ON MANIFOLDS ALBERTO S. CATTANEO

NOTES ON MANIFOLDS ALBERTO S. CATTANEO NOTES ON MANIFOLDS ALBERTO S. CATTANEO Contents 1. Introduction 2 2. Manifolds 2 2.1. Coordinates 6 2.2. Dimension 7 2.3. The implicit function theorem 7 3. Maps 8 3.1. The pullback 10 3.2. Submanifolds

More information

Solutions to Problem Set 5 for , Fall 2007

Solutions to Problem Set 5 for , Fall 2007 Solutions to Problem Set 5 for 18.101, Fall 2007 1 Exercise 1 Solution For the counterexample, let us consider M = (0, + ) and let us take V = on M. x Let W be the vector field on M that is identically

More information

MATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X.

MATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X. MATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X. Notation 2 A set can be described using set-builder notation. That is, a set can be described

More information

INVERSE FUNCTION THEOREM and SURFACES IN R n

INVERSE FUNCTION THEOREM and SURFACES IN R n INVERSE FUNCTION THEOREM and SURFACES IN R n Let f C k (U; R n ), with U R n open. Assume df(a) GL(R n ), where a U. The Inverse Function Theorem says there is an open neighborhood V U of a in R n so that

More information

KUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS

KUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS KUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU 1. Introduction These are notes to that show

More information

Metric Spaces Lecture 17

Metric Spaces Lecture 17 Metric Spaces Lecture 17 Homeomorphisms At the end of last lecture an example was given of a bijective continuous function f such that f 1 is not continuous. For another example, consider the sets T =

More information

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu

More information

Math General Topology Fall 2012 Homework 1 Solutions

Math General Topology Fall 2012 Homework 1 Solutions Math 535 - General Topology Fall 2012 Homework 1 Solutions Definition. Let V be a (real or complex) vector space. A norm on V is a function : V R satisfying: 1. Positivity: x 0 for all x V and moreover

More information

Part II. Riemann Surfaces. Year

Part II. Riemann Surfaces. Year Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised

More information

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA  address: Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework

More information

7.3 Singular Homology Groups

7.3 Singular Homology Groups 184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the

More information

Tangent bundles, vector fields

Tangent bundles, vector fields Location Department of Mathematical Sciences,, G5-109. Main Reference: [Lee]: J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer-Verlag, 2002. Homepage for the book,

More information

3 Hausdorff and Connected Spaces

3 Hausdorff and Connected Spaces 3 Hausdorff and Connected Spaces In this chapter we address the question of when two spaces are homeomorphic. This is done by examining two properties that are shared by any pair of homeomorphic spaces.

More information

Chapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves

Chapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction

More information

NOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY

NOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY NOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY 1. Closed and exact forms Let X be a n-manifold (not necessarily oriented), and let α be a k-form on X. We say that α is closed if dα = 0 and say

More information

DIFFERENTIAL GEOMETRY 1 PROBLEM SET 1 SOLUTIONS

DIFFERENTIAL GEOMETRY 1 PROBLEM SET 1 SOLUTIONS DIFFERENTIAL GEOMETRY PROBLEM SET SOLUTIONS Lee: -4,--5,-6,-7 Problem -4: If k is an integer between 0 and min m, n, show that the set of m n matrices whose rank is at least k is an open submanifold of

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

The theory of manifolds Lecture 2

The theory of manifolds Lecture 2 The theory of manifolds Lecture 2 Let X be a subset of R N, Y a subset of R n and f : X Y a continuous map. We recall Definition 1. f is a C map if for every p X, there exists a neighborhood, U p, of p

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

More information

Differential Geometry (preliminary draft) Massimiliano Mella

Differential Geometry (preliminary draft) Massimiliano Mella Differential Geometry (preliminary draft) Massimiliano Mella Introduction These notes are intended for an undergraduate level third year. It is a pleasure to thank C. Bisi for a carefull reading. CHAPTER

More information

Introduction to Computational Manifolds and Applications

Introduction to Computational Manifolds and Applications IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Introduction to Computational Manifolds and Applications Part 1 - Foundations Prof. Jean Gallier jean@cis.upenn.edu Department

More information

3 COUNTABILITY AND CONNECTEDNESS AXIOMS

3 COUNTABILITY AND CONNECTEDNESS AXIOMS 3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D = X. X is separable iff it contains a countable dense subset. X satisfies the first

More information

3. Prove or disprove: If a space X is second countable, then every open covering of X contains a countable subcollection covering X.

3. Prove or disprove: If a space X is second countable, then every open covering of X contains a countable subcollection covering X. Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM January 24, 2015 Examiners: Dr. M. Elhamdadi, Dr. M. Saito Instructions: For Ph.D. level, complete at least

More information

Geometry Qualifying Exam Notes

Geometry Qualifying Exam Notes Geometry Qualifying Exam Notes F 1 F 1 x 1 x n Definition: The Jacobian matrix of a map f : N M is.. F m F m x 1 x n square matrix, its determinant is called the Jacobian determinant.. When this is a Definition:

More information

NAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key

NAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)

More information

Lecture 2. Smooth functions and maps

Lecture 2. Smooth functions and maps Lecture 2. Smooth functions and maps 2.1 Definition of smooth maps Given a differentiable manifold, all questions of differentiability are to be reduced to questions about functions between Euclidean spaces,

More information

Math 215B: Solutions 1

Math 215B: Solutions 1 Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an

More information

MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005

MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005 MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005 1. True or False (22 points, 2 each) T or F Every set in R n is either open or closed

More information

Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3

Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3 Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability

More information

Real Projective Space: An Abstract Manifold

Real Projective Space: An Abstract Manifold Real Projective Space: An Abstract Manifold Cameron Krulewski, Math 132 Project I March 10, 2017 In this talk, we seek to generalize the concept of manifold and discuss abstract, or topological, manifolds.

More information

Math 320-2: Final Exam Practice Solutions Northwestern University, Winter 2015

Math 320-2: Final Exam Practice Solutions Northwestern University, Winter 2015 Math 30-: Final Exam Practice Solutions Northwestern University, Winter 015 1. Give an example of each of the following. No justification is needed. (a) A closed and bounded subset of C[0, 1] which is

More information

Lecture 4: Stabilization

Lecture 4: Stabilization Lecture 4: Stabilization There are many stabilization processes in topology, and often matters simplify in a stable limit. As a first example, consider the sequence of inclusions (4.1) S 0 S 1 S 2 S 3

More information

DEVELOPMENT OF MORSE THEORY

DEVELOPMENT OF MORSE THEORY DEVELOPMENT OF MORSE THEORY MATTHEW STEED Abstract. In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined

More information

Sanjay Mishra. Topology. Dr. Sanjay Mishra. A Profound Subtitle

Sanjay Mishra. Topology. Dr. Sanjay Mishra. A Profound Subtitle Topology A Profound Subtitle Dr. Copyright c 2017 Contents I General Topology 1 Compactness of Topological Space............................ 7 1.1 Introduction 7 1.2 Compact Space 7 1.2.1 Compact Space.................................................

More information

Real Analysis Chapter 4 Solutions Jonathan Conder

Real Analysis Chapter 4 Solutions Jonathan Conder 2. Let x, y X and suppose that x y. Then {x} c is open in the cofinite topology and contains y but not x. The cofinite topology on X is therefore T 1. Since X is infinite it contains two distinct points

More information

(iv) Whitney s condition B. Suppose S β S α. If two sequences (a k ) S α and (b k ) S β both converge to the same x S β then lim.

(iv) Whitney s condition B. Suppose S β S α. If two sequences (a k ) S α and (b k ) S β both converge to the same x S β then lim. 0.1. Stratified spaces. References are [7], [6], [3]. Singular spaces are naturally associated to many important mathematical objects (for example in representation theory). We are essentially interested

More information

NOTES FOR MATH 5520, SPRING Outline

NOTES FOR MATH 5520, SPRING Outline NOTES FOR MATH 5520, SPRING 2011 DOMINGO TOLEDO 1. Outline This will be a course on the topology and geometry of surfaces. This is a continuation of Math 4510, and we will often refer to the notes for

More information

Math 598 Feb 14, Geometry and Topology II Spring 2005, PSU

Math 598 Feb 14, Geometry and Topology II Spring 2005, PSU Math 598 Feb 14, 2005 1 Geometry and Topology II Spring 2005, PSU Lecture Notes 7 2.7 Smooth submanifolds Let N be a smooth manifold. We say that M N m is an n-dimensional smooth submanifold of N, provided

More information

An Introduction to Riemannian Geometry

An Introduction to Riemannian Geometry An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity Leonor Godinho and José Natário Lisbon, 2004 Contents Chapter 1. Differentiable Manifolds 3 1. Topological Manifolds

More information

ULTRAFILTER AND HINDMAN S THEOREM

ULTRAFILTER AND HINDMAN S THEOREM ULTRAFILTER AND HINDMAN S THEOREM GUANYU ZHOU Abstract. In this paper, we present various results of Ramsey Theory, including Schur s Theorem and Hindman s Theorem. With the focus on the proof of Hindman

More information

10. The subgroup subalgebra correspondence. Homogeneous spaces.

10. The subgroup subalgebra correspondence. Homogeneous spaces. 10. The subgroup subalgebra correspondence. Homogeneous spaces. 10.1. The concept of a Lie subgroup of a Lie group. We have seen that if G is a Lie group and H G a subgroup which is at the same time a

More information

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu

More information

Solutions to Tutorial 8 (Week 9)

Solutions to Tutorial 8 (Week 9) The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/

More information

An introduction to cobordism

An introduction to cobordism An introduction to cobordism Martin Vito Cruz 30 April 2004 1 Introduction Cobordism theory is the study of manifolds modulo the cobordism relation: two manifolds are considered the same if their disjoint

More information

Degree of circle maps and Sard s theorem.

Degree of circle maps and Sard s theorem. February 24, 2005 12-1 Degree of circle maps and Sard s theorem. We are moving toward a structure theory for maps certain maps of the interval and circle. There are certain fundamental notions that we

More information

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

More information

DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM

DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM DIFFERENTIAL TOPOLOGY AND THE POINCARÉ-HOPF THEOREM ARIEL HAFFTKA 1. Introduction In this paper we approach the topology of smooth manifolds using differential tools, as opposed to algebraic ones such

More information