Herman Brown 416 (my office)

(713) 348-5265 (my office) (713) 348-4829 (math office)

cochran@rice.edu

http://www.math.rice.edu/~cochran

**Teaching Assistant (will grade homeworks: **Ms. Taylor McNeill,

**For Homework Assignments**see bottom of this page or login to OWLSPACE

**Course Web Page:**
http://math.rice.edu/~cochran/teaching/Math454Fall2012/Math 444Fall2012webpage.html .

**OWLSPACE:** Owlspace login

**Prerequisites:** Math 443, Math 356 or equivalent. More specifically , some point set topology will be assumed: topological spaces, continutity, compactness, connectedness, metric spaces, quoitient spaces. A semester course in this material (e.g Math 443) would be ideal background, but the text has a good review of this material in the first four chapters. Some group theory will be used: groups, homomorphisms, normal subgroups, quotient groups. Please see me if you have further questions.

**Required Textbook : ** Introduction to Topological Manifolds, second edition, by John M. Lee, Springer.

**Overall course objectives and expected learning outcomes:** the fundamental paradigm of algebraic topology- what does it seek to accomplish/what does it accomplish, Quotient spaces,adjunction spaces, CW-complexes, topological classification of surfaces,Euler charcateristic, homotopy of maps and homotopy equivalence of spaces, the fundamental group of a space (definition and how to calculate it), covering spaces and relationship to fundamental group, applications,introduction to higher-homotopy groups, introduction to fiber bundles.

**Assessment:**This class will be taught as a first-year graduate class. Undergraduates who enroll in 444 will be assigned fewer homework problems and will be assigned grades separately. There will be a final exam (35%) and one mid-term exam (25%). Homework will count for 40% of the grade. Attendance will not be part of the grade, but the student is responsible for ALL information conveyed in class and all material assigned in the book. Some material will be available in class, but NOT on Owlspace. The student is also responsible for reading email messages from the professor via Owlspace and logging in to Owlspace sometimes for assignments. Students enrolled for Math 539 will be required to do additional homework problems compared to stuents in Math 444.

Good mathematical exposition will be counted on both exams and homework. Homework assignments will be assigned via Announcements at Owlspace and/or on this page. You are strongly encouraged to work together in small groups. However, you are not allowed to
use solutions from the internet, nor any solutions you might find in any written form. Moreover each
student must write up her/his own homework without copying anyone else's. Homework is due AT THE START of CLASS on the assigned day. Late homework will be accepted occasionally with good reason, but usually will be uncorrected and assigned a score equal to 60% of the average of the student's previous HW scores

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.

Information on this syllabus (except attendance policy) may change with reasonable notice given via email from Owlspace.

**Homework Assignments:**

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Due Date Assignment
Before Wednesday's class, READING: Chapter 1 (optional); Review Chapter 2 as needed, noting the definition of a manifold; Review Chapter 3 as needed, especially subspace topology, definition of a topological embedding, Lemma 3.23. On Wednesday we will begin by reviewing the quotient topology and proceed to Chapter 5
is available via the announcements section of the Owlspace site
Read p.42-44 (def. of manifold with boundary); Chapter 5: read "lightly" pages 127-132; I prefer the definition of CW-complex given as Theorem 5.20 in textbook and as given in class; read pages 134-135; SKIM pages 135-142 noting various topological properties, especially Thm. 5.14 and Corollary 5.15 which we will mention in class Monday.
** REQUIRED PROBLEMS DUE 8/31** 3-1 page 81, 3-14 and 3-15 (see Prop.3.56; only show Hausdorff for 3-14, not 3-15); 4.1 page 122 is available via the announcements section of the Owlspace site Problem set is available via the announcements section of the Owlspace site
Skim 178-181; Read very carefully 183-185
** REQUIRED PROBLEMS DUE 9/14** 6-2 page 181, 6-5, Exercise 7.6 and 7.8 page 187; 7.1 page 214; Prove that any continuous map f from a space X into the upper hemisphere of the n-sphere is homotopic to a constant map.
The lecture on Friday covered pages 202-205. Please read the defintion of contractible space carefully as I did not cover it in class. pages 206-208 are optional (it is good to know the statement of Prop. 7.46) ** REQUIRED PROBLEMS DUE 9/28: Exercise 7.42 page 202 except replace part c with "Some point of X is a deformation retract of X". This is mostly just an exercise in applying the definitions; Page 215 do 7-10,7-11,7-14. For those in the graduate version do 7-13 also.**