Math 141. Fall'18. Elementary Differential Topology (CCN: 24448)
Instructor: Alexander Givental
Lectures: TuTh 11-12:30, in 3 Evans
Office hours: Wed 4-6, in 701 Evans
Prerequisites: Math 104 and Math 110
Textbook: "Elementary differential topology" by Guillemin and Pollack, any edition (there are two different ones which seem identical inside)
Grading: 40% homework, 40% final, 20% mid-semester tests (I'm leaning toward having quizzes after each chapter instead of one midterm, but we'll see ...)
Homework: Will be posted weekly to this website, and due on Th of the next week in class.
HW1, due Th, Aug. 30: Read Sections 1,2 of Chapter 1, as well as
the formulations of all exercises, and think whether you understand how to
solve them. Submit solutions of:
1. Show the equivalence of two competing definitions of smooth maps
f: X --> Y between manifolds: (i) it is a map which is smooth in the sense
of the definition of smooth maps between subsets of Euclidean spaces
(on p. 2 of the textbook), and (ii) when expressed in local coordinated near
any x in X and near f(x) in Y, it is given by smooth functions.
2. Often the sets of solution to some systems of non-linear equations are manifolds. To guarantee this, one needs to check that the hypotheisis of the Implicit Function Theorem are fulfilled. Temporarily suspending this check, establish a diffeomorphism between the following sets X and Y (which are
in fact manifolds, one in R^9, the other in R^6): X = the set of 3x3 matrices U satisfying U^tU=I (where ^t means transposition, I identity) and det U=1;
Y = the set of triples (not pairs -- there was an error in the formulation, after all) of complex numbers (z,u,w) satisfying z^2+u^2+w^2=0, |z|^2+|u|^2+|w|^2=1. Make an educated guess on whether they are also diffeomorphic to RP^3, the
manifold (in the abstract sense) of 1-dimensional subspaces in R^4
(= S^3 with all pairs of diametrically opposite points identified).
3. Ecercise 10 from Section 1.
4. Exercise 18 from Section 1 (note a typo in (b)).
5. Find an error (not a typo) in one of the exercises of Section 1.
Answers to HW1
HW2, due Th, Sep. 6: Read sections 3,4 of Chapter 1. Solve:
1. The goal of this exercise is to convince you that on a manifold,
the differential of a smooth function is well-defined, but such a notion of "math 53" as the gradient doesn't make sense. Namely. Let f: R^2 --> R be a smooth function in two variables x, y, and dx, dy denote the differentials of x and y (which are smooth functions, after all). Now make the change of variables x=u+v, y=v to define function g(u,v):=f(u+v, v).
(a) Take the differential df=f_x dx+ f_y dy, apply to it our change of the variables, and compare the result with dg = g_u du+g_v dv.
(b) Now take the gradient of f (which is the vector with components f_x,f_y on the (x,y)-plane), and transform it to the (u,v)-coordinates. Compare the result with the gradient of g (which is the vector with components g_u, g_v on the (u,v)-plane.
2. Let G: (r,\th) |--> (x,y)=(r \cos \th, r \sin \th) be the polar change of coordinates on the plane. Consider the tangent vector d/d\th at the point (r,\th) ("d" in fact should be "partial d", but I don't have this font), and compute its image under the differential dG at that point. [The answer should have the form a d/dx + bd/dy of a tangent vector at the corresponding point (x,y).]
3. Exercise 4 of section 3.
4. Consider the parametric curve R --> R^3: t |--> (t^2,t^3,t^4). (a) Find all points where it is not an immersion, and sketch the image of it. (b) Describe the image by two equations: f(x,y,z)=0, g(x,y,z)=0. Find all points in where your map R^3-->R^2 given by the functions f,g is not a submersion.
5. Ecexrcise 10 from Section 4.
Answers to HW2
On Tue, Sep. 11, we'll have quiz on compactness!
HW3, due Th., Sep. 13: Read sections 5,6,7 of Chapter 1, as well as all Exercises in Sections 5-6, and make sure you know how do solve them - they all are useful (although many may look mundane or even trivial). Submit solutions of Exercises 3, 8, 10 from Section 5, and 7, 9 from Section 6.
Answers to HW3
HW4, due Th., Sep. 20: Read Chapter 1 to the end plus Appendix 1. Solve:
Exercises 8,9,12 from Section 8, as well as the following two problems:
A.
Consider function f(x,y)=x^2y+y^2-ax-by in R^2, depending on two parameters
a and b. On the plane of parameters, drow the locus (a curve) for which f is not a Morse funtion. For ech connected component of the complement to this curve, find how many critical points the function has, and how many of them are local maxima, local minima, and saddles.
B. On the sphere x^2+y^2+z^2=1 in R^3, consider function ax^2+by^2+cz^2, and assume a>b>c. Show that it is a Morse function, find its critical points, and for each of them determine whether it is a local maximum, local minimum, or a saddle.
Answers to HW4
HW5, due Th., Ser. 27: Read Sections 1,2 of Chapter 2. Solve:
1. Verify by computation in local coordinates that the commutator of two vector fields is a vector field.
2. Let A,B: \R^n --> \R^n be two linear maps, and v_A, v_B corresponding linear vector fields (defined
by v_A (x) = Ax). Verify that [v_A,v_B] is the linear vector field corresponding to the commutator [A,B]=AB-BA of the linear maps.
3. Verify that in the 3-dimensional space of anti-symmetric 3x3-matrices, the operation defined as the commutator of matrices is identical to the cross-product operation with 3-dimensional vectors.
4. On the line, consider 3 vector fields: d/dx, xd/dx, and x^2 d/dx. Check that they span a 3-dimensional Lie subalgebra in the Lie algebra of all vector fields on the line. To find out what 3-dimensional group of transformations on the line they generate, integrate these vector filds (i.e. solve differential equations: dx/dt =1, dx/dt=x, and dx/dt=x^2), and consider all compositions of the diffeomorphisms defined by these solutions.
5. Let v_A be a linear vector field on \R^n corresponding to a linear map A: \R^n-->\R^n (as in problem 2). Show that the flow of this vector field (i.e. the 1-parametric group of diffeomorphisms defined by the general solution of the differential equation dx/dt = Ax) is given by the matrix exponential funcion: x(t)=e^{tA}x(0) (that's straightforward). Now apply these diffeomorphisms (carefully, the way diffeomorphisms are supposed to act on vector fields of tangent vectors) to the linear vector field v_B corresponding to another linear map B. You'll get a family of vector fields depending on t, and now compute the derivative in t at t=0, and express the result in temrs of the commutator [v_A,v_B].
Answers to HW5
HW6, due Th., Oct. 4: Read Sections 1-2 of Chapter 2. Solve:
Problem 10 of Section 1, Problems 2 and 5 of section 2, and the following problems on Lie groups:
A. SU_2 x SU_2 /{(I,I),(-I,-I)} is isomorphic to SO_4.
B. Show that over complex numbers, the 3-dimensional Lie algebra of vector fields (a+bx+cx^2) d/dx on the line is isomorphic to the cross-product Lie algebra of vectors in \R^3.Give two proofs: one by explicit formulas, another computation-free, based on a geometrical relationship between the corresponding Lie groups.
Answers to HW6
HW7, due Th, Oct. 11: Read Chapter 2 to the end. Solve:
A. Prove that the Inverse Function Theorem is equivalent to the following Implicit Function Theorem: Given a smooth map F: R^n x R^m --> R^m, (x,y) |--> z=F(x,y), such that F(0,0)=0, and det [ dz/dy ](0,0) is non-zero. Then in a neighborhod of x=0 there is a unique function y=f(x), f: R^n --> R^m, such that f(0)=0 and F(x,f(x))=0 identically. In other words, the equation F(x,y)=0 locally uniquely defines y as a function of x, or in yet other words, the solution set to F(x,y)=0 is locally the graph of a smooth function x |--> y=f(x).
B. Prove (using intersection theory mod 2) that any smooth vector field on RP^{2n} has zeroes. Give a counter-example on RP^{2n-1}.
C. Compute mod-2 intersection indices of subspaces RP^n and RP^m in RP^{n+m}, and derive that spheres and projective spaces of the same dimension are not diffeomorphic.
Solve exercise 4 from Section 3 and exercise 13 from Section 4 of Chapter 2.
Answers to HW7
HW8, due Th, Oct. 18: Read sections 1-5 of Chapter 3. Read all exercises in secions 2,3, and solve them (most will look trivial after our in-class discussions, but it is a good test of your understanding). Solve exercises 13, 23 of section 3.2 and 2,19 of section 3.3, as well as
Problem (*): Show that the Gram-Schmidt orthogonalization algorithm can be used to provide a deformational retraction of GL_n(R) to O_n (i.e. a homotopy identical on O_n and contracting GL_n(R) to O_n), and then show that O_n (and hence GL_n(R)) has exactly two connected components.
Answers to HW8
On Tue, Oct. 23, we have Quiz 2 on Orientations
HW9, due Th, Oct. 25: Read Chapter 3 to the very end. Solve as many exercises as you can. Submit solutions of:
1. Exercise 10 of Section 3.
2. Prove that the Euler characteristic is well-defined even for non-orientable manifolds, and that it is still a diffeomorphism invariant.
3. Compute the Euler charactristic of RP^n for all n=0,1,2,...
4. Exercise 10 of Section 4.
5. Exercise 18 of Section 5.
Answers to HW9
HW10, due Th, Nov. 1: Here is a reference for determinants. Read Sections 1-3 of Chapter 4. Alternatively, read sections 32-34 from Arnold's book. Solve:
1. Prove Binet-Cauchy's formula: Given kxn-matrix A and nxk-matrix B,
det AB = \sum_I (det A_I) (det B_I) where A_I (resp. B_I) is the kxk-matrix formed by k columns of A (resp. rows of B) specified by the multi-index
I=(i_1<...< i_k).
2. Prove Laplace's theorem (cofactor expansion with respect to several rows/columns): Given an nxn-matrix A, det A = \sum_I (-1)^|I| (\det A_I) (det A'_I) where A_I is the kxk-matrix on the intersection of the first k rows of A and the columns speified by the multi-index I=(i_1<...< i_k), A'_I is the (n-k)x(n-k) matrix on the intersection of the last n-k rows and the remaining n-k columns of A, and |I|=1+...+k+i_1+...+i_k.
Hint to both problems: Use our description of all poly-linear totally anti-symmetric functions of k-tuples of n-dimensional vectors.
3. Let v be a non-zero vector in a vector space V. Define the contraction map i_v from the space of exterior k+1-forms on V to the space of exterior k-forms on V by (i_v F)(v_1,...,v_k):=F(v, v_1,...,v_k). Prove that the kernel of i_v coincides with the image of the previous i_v (taking a k+2-form into a k+1-form).
4 - 5: Solve exercises 2 and 6 from Section 2 of Chapter 4.
Answers to HW10
HW11, due Th, Nov. 8: Read Sections 5, 4, 7 of Chapter 4 and/or Sections 35-36 from Arnold's book. Solve:
1. Prove that an n-dimensional manifold is orientable if and only if it has a non-vanishing differential n-form. (Such a form is called a volume form on the manifold.)
2. The differential 1-form A:=du-p_1dq_1-p_2dq_2-...-p_ndq_n is called contact form on the space \R^{2n+1} with coordinates u,p_1,...,p_n,q_1,...,q_n. Compute A \wedge dA \wedge ... \wedge dA (exterior product of A with n copies of dA) and show that it is a volume form on \R^{2n+1}.
3. A differntial form F is called closed if dF =0. Show that a differntial 1-form F is closed if and only if L_u F(v) - L_v F(u) = F([u,v]) for any vector fields u, v. (Here L_u is the derivative of a function in the direction of u - what we once also denoted by D_u).
4. Let F=Pdx+Qdy be a differential 1-form in \R^2, and g: \R^2 --> \R^2 : (u,v)|--> (x(u,v),y(u,v)) a smooth map. Check by direct computation that g^*dF=d(g^*F).
5. Let w=A d/dx + B d/dy + C d/dz be a vector field in \R^3, V=dx \wedge dy \wedge dz the standard volume form, and s: \R^2 --> \R^3: (u,v) |--> (x(u,v),y(u,v),z(u,v)) a smooth parametric surface. Show that the integral of s^*(i_w V) (which is a differential 2-form in \R^2) over any domain D in \R^2 is equal to the flux of the vector field w across the piece of the surface parameterized by D.
Answers to HW11
HW12, due Th., Nov. 15: Read Sections 6 and 8 of Chapter 4. Solve:
1. Prove that the value of the integral \int_c w , where w is a closed p-form (i.e. dw=0) and c is a p-dimensional cycle (i.e. dc=0), doesn't change if an exact form (i.e. d(something)) is added to w and/or a boundary chain (i.e. d(someting)) is added to c.
2. Prove that on a compact oriented n-dimensional manifold X, the value (a,b):= \int_X a \wedge b, where a is a closed p-form, and b is a closed (n-p)-form, doesn't change if an exact form is added to a or/and b.
3. Physicist call a vector field in \R^{2n} with coordinates (p_1, q_1,...,p_n, q_n) hamiltonian if it has the form dp_i/dt = - H_{q_i}, dq_i/dt = H_{p_i}, i=1,...,n, where H(p,q) is a smooth function (called the hamitonian ). Show that a vector field in \R^{2n} is hamiltonian if and only if its flow preserves the following closed differential 2-form (called symplectic ): dp_1\wedge dq_1+...+dp_n\wedge dq_n. ( Hint: Use Cartan's homotopy formula.)
4. Let V = dx_1\wedge ... \wedge dx_n be the standard volume form in \R^n, E = x_1 d/dx_1+...+x_n d/dx_n the Euler vector field (equal x at a point x), and |x|=(x_1^2+...+x_n^2)^{1/2} the distance to the origin. Compute\int_{d B} |x|^{-n} i_E V, where B is the cube |x_i|=<1, i=1,..., n. ( Hint: First show that the differential (n-1)-form is closed, and then -- be cautious!)
5. Compute de Rham cohomology of S^1.
Answers to HW12
HW13, due Th., Nov. 29: Read Chapter 4 to the end. Solve:
1. Assume that the flow g^t of a vector field v on a manifold X defines a circle action, i.e. g^{t+1}=g^t for all t. Following the logic from the proof of Poincare's lemma, show that for any closed differential form w on X, (g^t)*w-w is exact, and derive from this that the circle average \int_0^1 (g^t)^* w dt represents the same cohomology class as w.
2. Compute the De Rham cohomology of the n-dimensional torus T^n=S^1 x ... x S^1. Namely, in the De Rham complex of differential forms on T^n, consider the sub-complex of all differential forms invariant under translations on the torus, and prove that the inclusion of this subcomplex into the whole complex induces an isomorphism in cohomology.
3. Compute cohomology of the De Rham complex of compactly supported differential forms on \R^n. ( Hint: Read the end of section 4.6 including exercises 9 and 10.)
4. Prove that integration of differential n-forms over compact connected oriented n-dimensional manifold X identifies the top De Rham cohomology H^n(X) with \R. ( Hint: See Exercise 7 in Section 8.)
5. Solve Exercise 8 of Section 9.
Answers to HW13 On Tue, Nov. 27, let's have a 5-min quiz on manipulation with differential forms
Solutions to the final exam