Math 444: Geometric Topology

MWF 9AM, HB 423

Professor Tim Cochran
Herman Brown 416 (my office)
(713) 348-5265 (my office) (713) 348-4829 (math office)
cochran@math.rice.edu
http://www.math.rice.edu/~cochran
Office hours: Tuesday 2:30-4:00, Thursday 2:30-3:30 and by appointment

Grader: Andrew Elliott, elflord@rice.edu

Course Web Page: http://math.rice.edu/~cochran/teaching/Math444Fall07/Math444Fall07.html .

OWLSPACE: Owlspace login


Prerequisites:

Textbook : Introduction to Topological Manifolds by John M. Lee, Springer. See Lee errata for corrections


Grading:



Any student with a documented disability needing academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain confidential. Students with disabilities will need to also contact Disability Support Services in the Ley Student Center.


Homework Assignments:
#1. Read pages 17-29 (should be review) and read 30-35 before Wednesday's class
#2. For Friday Read pages 39-51. Due next Wednesday in class: 2-10,3-2,3-6,3-8 (ignore second countable),3-10, 3-11 (just show locally Euclidean part).
#3. Before Wednesday Read pages 52-61
#4. Due Friday in class. Read pages 91-101; D0 page 63 3-14, page 114 5-2
#5. Read pages 102-109 Due next Wednesday in class: Exercise 5.4 b page 95; Prove Lemma 5.4 b,d ; 5-12 page 115 only for 2-manifold and only for a single barycentric subdivision (see middle of Figure 5.12; Prove that the space obtained from a 2-disk by identifying its boundary to one point is homeomorphic to the two sphere (subspace topology from 3-space) (use Corollary 3.30 and Lemma 4.25 or use Corollary 3.32)
#6. Read pages 117-138. Due next Wednesday 9/26 in class: 1. Prove that a submanifold of an orientable n-manifold is orientable (Since we have only really defined orientability for a triangulated manifold, you cannot really prove this but you should prove that if K is a simplicial complex consisting entirely of some n simplices and all of their faces, in which each (n-1) simplex is the face of at most two n simplices, and L is a subcomplex of K with this same property, then if K is orientable so is L. Your proof should be short.)2. Without using the classification theorem, prove that a 2-manifold is non-orientable if and only if it contains an embedded Moebius band (once again you cannot really prove this but you can prove some ``moral'' equivalent. Assume that all Moebius bands are triangulated as a string of k triangles (k variable) in the most obvious way possible (see Figure 5.7) and assume that any embedding of a Moebius band in a triangulated surface S=|K| is the underlying space of a subcomplex L that has one of these triangulations). Do 6-3, 6-4 page 146. After class on Monday do 6-1 page 146.
#7. For Friday 9/28: Read pages 139-145.
#8. For Monday 10/1 Read pages 147-152
#9. For Wednesday 10/3 Read pages 153-158; Do: #E1.Derive a formula for the Euler characteristic of a connected sum of two n-dimensional manifolds in terms of the Euler characteristics of the manifolds. Do: Exercise 7.1 page 151; Exercises 7.2.7.3 on page 156; also on page 176 do problem 7-1. #E2. A space X is Contractible if the identity map X---X is homotopic to a constant map. Show that a contractible space is path-connected. Show that any continuous map f:Y--X where X is contractible, is homotopic to a constant map.
#10 For Monday 10/8: Read pp. 158-166
#11. For Wednesday 10/10 Do page 176 #7-2d, #7-4, #7-5, page 162 Exercise 7.7 (use definitions)(do a implies b implies c implies a),E#1: Let x,y be distinct points of a simply-connected space X. Prove that there is a unique path homotopy class of paths from x to y., E#2. Let X be a topological space that is the union of a countably infinite number of path-connected spaces X_n containing the base point q. Assume these are nested (X_n contained in X_n+1) and that for any compact set A of X there exists an integer n such that A is contained in X_n. Prove that for any class [f] in pi_1(X) there exists some n such that [f] is in the image of the map i_n_*:\pi_1(X_n)--\pi_1(X) induced by inclusion.
#12. For Friday 10/20 Read 167-170 Optional 170-173 (you will learn about this in Math 445)
Do: p.177 #7-13 (sorry I forgot to talk about trees- will do this On Wednesday-but it's easy stuff); Use the specific definitions from class for the following: #E1. Suppose A is a strong deformation retract of X. Prove that, for each n, the inclusion map induces an isomorphism pi_n(A)--pi_n(X) (basepoint in A). Use this and Proposition 7.30 to prove that any two spaces that are homotopy equivalent have isomorphic pi_n #E2. Prove that an element [f] in pi_n(X) is zero if and only if the associated map of the sphere f:S^n---X extends to a continuous map of the n+1 ball into X. #E3. Assume that pi_m(S^n) is 0 if m is less than n and is non-zero if m equals n. Prove that R^p is NOT homeomorphic to R^q if p is not equal to q. Be careful since these ARE homotopy equivalent). #E4. Assume pi_1(S^1) is infinite cyclic. Observe that for a wedge of m circles X there are m circles each of which is a (subspace of and) retract of X. Prove that the corresponding subgroups of pi_1(X) intersect only in the zero element. Prove that there is an epimorphism from pi_1(X) to the direct sum of m copies of the infinite cyclic group Z.
#13. For Wednesday 10/24 Read pp.179-190. There is a gap in the proof of Theorem 8.7. Also there are eratta for problems on 191-192: Page 191, Problem 8-7: In the third line of the problem, change '( ) to '( ). • Page 192, line 4: Change the definition of ' to '(x) = (x - f(x))/|x - f(x)|;
Do: page 191-192 8-1, 8-2, E#1: Prove that a p-manifold is not a q-manifold unless p=q. Try to use strategy of 8-4 and 8-5 and use one of exercises from our last Homework. 8-7, 8-8, 8-9 (these are some cool results!) Those of you who have not had group theory- now is a good time to review appendix on group theory.
#14. For Monday 11/5 Read 193-207 and 209-212.
#15. For Wednesday 11/7; Do pg. 208 9-1, 9-2, 9-3, and page 230 10-6, 10-7
. #16. For Monday 11/12 Read 201-229.
#17. For Wednesday 11/14 Do page 230 # 10-1,10-2, 10-5,10-9,10-12, Suppose f:S^n-1--Y is a continuous map from the n-1 sphere to a space Y. Consider the space Y disjoint union B^n and let X be the identification space (quotient topology) obtained from this disjoint union by identifying x~f(x) for each x in the boundary of B^n, which is S^n-1. X is called the result of attaching an n-cell to Y via the map f. Find a formula that gives the effect of attaching an n-cell on pi_1 (different for different n). After you are finished look at problem 10-8.
#18. For Friday 11/16 Read pp. 233-238
#19. For Monday 11/19 Read 239-242
#20 For Wednesday 11/21 Do: page 235 Exercise 11.2, page 252 # 11-2,11-3b (use Ex. 11.2),11-4,11-6,11-7,11-8 (first show any compact non-orientable surface without boundary is homeomorphic to connected sum of tori and ONE copy of RP(2)).
#21 For Wednesday 11/28 Read rest of Chapter 11 Do: page 252 # 11-13,11-14,11-15; with regards to the hand-out from class last Wednesday answer the following questions: E1. Which of the covering spaces in #1-11 are regular (normal) covering spaces ? E2. Note that the covering spaces indicated in (3) and (4) are identical but show that the induced subgroups using the two indicated basepoints are not identical. The general theory says that they must be conjugate subgroups- what is the conjugating element (using the theory)? E3. What are the groups of covering transformations in each case? REMARK: Note that 5) and 6) are distinct covering spaces (one can show the subgroups are not conjugate but you need not do it) but since the spaces X^~ are the same, the induced subgroups are ISOMORPHIC since each is isomorphic to the free group of rank 4.
#21 For Friday 12/7: Last Problem set
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