Charles P. Boyer, Math 321
Math 536: Homework
Assignment 1:
Exercise #1: Prove that the Grassmannian G(k,n) is locally Euclidean; #2: Lee, pg 20, Ex. 1.47; #3: Lee, pg 20, Ex. 1.48.
Due 2/09/11
Assignment 2:
Exercise #1: If $M_1$ and $M_2$ are smooth manifolds, show that the product space $M_1\times M_2$ is a smooth manifold. #2: Lee, pg. 23, Ex. 1.56; #3: Lee, pg. 26, Ex. 1.64.
Due 2/16/11 by 5pm.
Assignment 3:
Exercise #1: First exercise on page 3.35 of notes3. #2: Prove theorem 3.6 of notes3. #3: Prove theorem 3.7 of notes3. Due 2/23/11 by 5pm.
Assignment 4:
Exercise #1: Second exercise on page 3.49 of notes3 (about the real symplectic group Sp(n,R)\subset GL(2n,R)). #2: Let G be a Lie group acting smoothly on a smooth manifold M. Show that a G-orbit is an immersed submanifold of M. #3: First exercise on page 3.60 of notes4. Due 3/02/11 by 5pm.
Assignment 5:
Exercise #1: Do the first two exercises of notes5; (a): show that a smooth covering map is a submersion; (b) show that a deck transformation maps a fiber of the covering map to itself, and that if U is an evenly covered open set such that $\pi^{-1}(U)=\sqcup_i U_i$ then either $hU_i\cap U_j=\emptyset$ or $hU_i=U_j$. Exercise #2: (exercise 9.5 of John Lee's book, pg 224) Let G be a discrete group acting smoothly on a smooth manifold M. Show that the action $(g,p)\mapsto gp$ is proper (that is the map $A:G\times M\rightarrow M\times M$ defined by $A(g,p)=(gp,p)$ is a proper map) if and only if the following two conditions hold:
(i) Each p in M has a neighborhood U such that $gU\cap U=\emptyset$ for all but a finite number of $g\in G$.
(ii) If the points p and q of M are not on the same G-orbt, then there exist neighborhoods U of p and V of q such $gU\cap V=\emptyset$ for all $g\in G$.
Exercise #3: Prove Proposition 4.2 of notes6, that is prove that the set of germs $C^\infty(p)$ of smooth functions at a point p of M forms an associative commutative algebra with unit. Due 3/9/11.
Assignment 6:
Exercise #1: Do the first exercise on page 4.14 of notes6 (involving dimension of image of differential plus dimension of its kernel). Exercise #2: Do the last exercise on page 4.18 of notes6 (Enneper's surface). Exercise #3: Let $N_1$ and $N_2$ be two regular (imbedded) submanifolds of a smooth manifold $M$. Show that if $N_1$ and $N_2$ intersect transversely, then the intersection $N_1\cap N_2$ of $N_1$ and $N_2$ is a regular submanifold of $M$. What is the dimension of $N_1\cap N_2$? Due 3/30/11.
Assignment 7:
Exercise #1: Prove theorem 4.5 of notes7 (pg. 4.27). Exercise #2: exercise on page 4.29 of notes7. Exercise #3: the second exercise (about tangent bundle of product manifolds) on page 4.30 of notes7. Due 4/6/11.
Assignment 8:
Exercise #1: The exercise on page 4.49 of notes8. Exercise #2: The exercise on page 4.50 of notes8, do parts 3,4,5, and 6. Due 4/13/11.
Assignment 9:
Exercise #1: Prove part 4) of Lemma 4.4 on page 4.59 of notes8. Exercise #2: The first exercise on page 4.65 of notes8 (about turning an associative algebra into a Lie algebra). Exercise 3: The third exercise on page 4.66 of notes8 (about Lie algebra structure constants). Due 4/20/11.
Assignment 10:
Exercise #1: The first exercise on page 4.66 of notes8 (F-related vector fields and Lie algebra homomorphism). Exercise #2: The exercise on page 6.2 of notes9 (about the induced bundle). Exercise 3: The second exercise on page 6.9 of notes9 (identify tensor fields of type (1,1) with endomorphisms of the tangent bundle). Due 4/27/11.
Assignment 11:
Exercise #1: Do the exercise on page 6.109 of notes11. Exercise #2: Do the exercise following Proposition 6.5 of notes10 (about pullbacks on the symmetric tensor algebra). Exercise #3: The exercise on page 6.70 of notes10 (about the cup product on de Rham cohomology). Due 5/4/11.