HOMEWORK

Assignment 1

Problem 1. Prove the excision theorem in singular cohomology.
Problem 2. Prove that S^n is not homeomorphic to any proper subspace of itself.
Problems 3-7. Exercises (18.10), (18,11), (18.13), (18.14) (note the symbol H# denotes reduced homology see page 48), (18.16) page 110-111, Greenberg and Harper.
Problem 8. Exercise (23.15), page 183, Greenberg and Harper.

Assignment 2

Problem 1. Compute cohomology of S^n with coefficients in R.
Problem 2. Compute the cohomology ring of S^n. Show for n>0, it is isomorphic to the polynomial ring R[X] mod the ideal X^2 where X is an indeterminate given degree equal to n.
Problem 3. If X is path connected and nonempty, prove H^0(X,R) is isomorphic to R.
Problem 4. If the path components of X are {X_i} prove H^n(X,R) is isomorphic to the direct product of the H^n(X_i,R).
Problem 5. Consider a group G with two generators a and b and one relation bab(a^(-1)). Consider a group H with two generators c and d and one relation (d^2)(c^2). Show they are isomorphic groups.
Problem 6. Calculate the fundamental group of S^n minus r points when n is at least 2 (r is a positive integer).
Problem 7. Let X be path connected. Let W union {V_i}i in I be an open cover of X by path connected sets containing the base point x_0.Assume W is simply connected and is a proper subset of every V_i. Assume in addition that for each pair of distinct indices, i and j, V_i intersection V_j is path connected. Prove that the fundamental group of X at x_0 is the free product of the fundamental groups of the V_i at x_0.

Assignment 3

Problem 1. Aubin, p.39. Problem 1.44 a, c
Problem 2 Aubin, p.39 Problem 1.45 a,b,c,d
Problem 3. Let R be the real line with the differentiable structure given by the maximal atlas determined by the atlas with the single chart (R, identity) (i.e, the usual differentiable structure). Let R' be the real line with the differentiable structure given by the maximal atlas determined by the atlas with the single chart (R, x goes to x^(1/3)). a). Show these differentiable manifolds are distinct. b). Show that the identity map from R to R' is NOT smooth. c). Show that R and R' are diffeomorphic smooth manifolds.
Problem 4, Let p be the usual covering map from R to S^1. Show that p is a local diffeomorphism.
Problem 5. Let f be a local diffeomorphism from R to R (again R is the real line), a). Prove that the image of f is a (possibly infinite) open interval in R. b) Prove that f is a (global) diffeomorphism from R to R. c) Construct a diffeomorphism of R onto a finite open interval.
Problem 6. Construct a local diffeomorphism of R^2 to R^2 which is not a diffeomorphism onto the image. (one approach uses Problem 4 above).
Problem 7. Prove real projective n-space is a compact manifold
Problem 8. Prove that a function f on the smooth manifold X is C^(infinity) if and only if for every chart (U, phi) in the atlas for X, f composed with (phi)^(-1) is C^(infinity) on phi(U),

Assignment 4

Problem 1. Prove that S^2 is diffeomorphic to the complex projective line as C^(infinity) manifolds). Prove that the Hopf map from S^3 to S^2 via (z_1, z_2) goes to [z_1: z_2] where z_1 and z_2 denote complex numbers whose complex absolute values squared add up to 1.
Problem 2. Prove that if X is a smooth manifold, Y a regular submanifold, P a point on Y, then any C^(infinity) function on Y at P has an extension to a C^(infinity) function on X at P.
Problem 3. Prove that O(2) the subgroup of GL_2 consisting of 2 by 2 real orthogonal matrices are a C^(infinity) manifold. Prove they form a Lie group under the multiplication of matrices.
Problem 4. Let F map R^2 to R^3 via (x,y) goes to (u,v,w)=(x,y,xy). Compute the differential of F acting on "partial differentiation wrt x" in terms of "partial differentiations wrt u, v, and w".
Problem 5. If M and N are smooth manifolds, let p_1 and p_2 be the two projections. For (x,z) in the product M X N, prove that the differential of p_1 x p_2 at (x,z) maps T_(x,z)(M X N) isomorphically onto T_x(M) X T_z(N).
Problem 6. Let G be a Lie group with multiplication map m and inverse map i. Show that the differential of m at (e,e) is addition in T_e(G) (use the identification in problem 5). Prove that the differential of i at e is the map to additive inverse, i.e. takes a point-derivation at e, say X_e to -X_e.

Assignment 5

Problem 1. Aubin, page 39, 1.44 b,d
Problem 2. Aubin, page 62, 2.42
Problem 3. Aubin, page 38, 1.37
Problem 4. a) If X is compact, Y connected, F a submersion from X to Y, then F is surjective. b) There are no submersions of compact manifolds into Euclidean spaces.
Problem 5. a) If f and g are immersions, then so is fXg. b) If f and g are immersions, then so is f composed with g. c) If f is an immersion then its restriction to any submanifold of its domain is also an immersion. d) When dim X = dim Y, f a smooth map from X to Y is an immersion if and only if f is a local diffeomorphism
Problem 6. Verify that the tangent space to the orthogonal group O(n) at the Identity is the vector space of skew-symmetric n by n matrices.
Problem 7. Prove that the set of mXn matrices of rank r is a submanifold of R^mn of dimension mn-(m-r)(n-r).

Assignment 6

Problem 1. Let M be an n-dimensional smooth manifold. For each P in M, let T_P(M)* denote the cotangent space of M at P, i.e., the dual real vector space to the tangent space T_P(M) of M at P. Define the cotangent bundle in an entirely analogous manner to our definition of the tangent bundle and prove in detail that the cotangent bundle is a smooth manifold.
Problem 2. Determine the critical points for the function f(x,y) = (xy)^2 from R^2 to R. Determine the values of (a,b) such that f + ax +by is a Morse function.
Problem 3. Let M be a smooth manifold. Let (U,f) and (V, g) be two charts on M. Let (T(U), f~) and (T(V), g~) be the corresponding charts on T(M). Compute the determinant of the transition map g composed with f^(-1) acting on T(U intersection with V).
Problem 4. For the map f(x, y) = (x^2 + y^2)exp(x^2 - y^2) find the critical points and classify as local max, local min or saddle point.

Assignment 7

Problem 1. Let M be a smooth manifold. Let I denote the closed unit interval. Let F be a continuous map from I X M into N, such that for every t in I, F_t is a smooth map of M into N. If F_0 is an immersion then there is a positive real epsilon such that every F_t, with t less than epsilon is an immersion.
Problem 2. A manifold whose tangent bundle is trivial is said to be parallelizable. If M is a manifold of dimension n, show that the parallelizability of M is equivalent to the existence of a smooth frame X_1,...X_n on M.
Problem 3. Exercise 2.41, Aubin.
Problem 4. Exercise 1.46, Aubin.
Problem 5. Exercise 3.27a,b,c. Aubin

Assignment 8

Problem 1. Suppose w is a smooth differential k-form on an open subset U of a smooth manifold. Let P be a point of M. Show there is a global k-form which agrees with w in a neighborhood of P.

Problem 2. Let x_1 and x_2 be the standard coordinates on R^2. Let D_i denote the vector field on R^2 defined by derivation wrt x_i. Let X_1 = (x_1)D_1 + (x_2)D_2 and X_2 = -(x_2)D_1 + (x_1)D_2. Find a smooth 1-form w on R^2 -{origin} such that w(X_1) = 1 and w(X_2) = 0.

Problem 3. If M is a manifold and N a regular submanifold then the inclusion i of N into M is smooth. As such we can pull back differential forms on M to differential forms on N. Consider the manifold M = R^3 with standard coordinates x, y, and z. Let N be the submanifold defined by ax + by = 0. Show that dx(wedge)dy is 0 on N.

Problem 4. Let M = R^2n with standard coordinates x_i. Let w be the 2 form which is the sum of dx_(2i-1)(wedge)dx_(2i) from 1 to n. Compute w^n (in the exterior algebra).

Assignment 9

Problem 1. If p is a smooth surjective submersion from N to M, prove that the pullback of p mapping the algebra of smooth forms on M to the algebra of smooth forms on N is an injective algebra homomorphism.

Problem 2. Let F map R^2 to R^2 via F(x,y) = (u,v) = ((x)^2 + ((y))^2, xy). Compute the pullback along F of udu + vdv.

Problem 3. Prove that if w is a smooth 1-form on a smooth manifold M and X and Y are smooth vector fields on M then (dw)(X,Y) = Xw(Y) - Yw(X)- w([X,Y]).

Problem 4.Let f be a smooth function on R^n and assume 0 is a regular value of f. Recall that the zero set M of f is an n-1 dimensional regular submanifold of R^n. Construct a nowhere-vanishing 1 form on M.

Problem 5.Prove any regular level set of a smooth real valued function f on R^n is an orientable n-1 dimensional smooth manifold.