Math 741, AB 210,MWF 9:00 to 9:50 am
Swatee Naik, Fall 2007

Office: AB 601D, tel 682 7175, fax 784 6378, email naik at unr.edu
Fall 2007 Office Hours: Wednesdays 1-2:15 and by appointment

Course Prerequisites:
MATH 440 (Point-Set Topology), MATH 331 (Abstract Algebra)

Course webpage: http://unr.edu/homepage/naik/classes/741/

Text book: Hatcher Allen, Algebraic Topology Chapters 0 and 2, Online

Reference book: Munkres, J.R., Algebraic Topology, Chapters 1-4


Final homework (Due December 7, 2007)
Give an example of a space which is connected but not path-connected.
Page 19: 12, 17, 18
Page 131: 2, 11, 15, 22
Page 135: 4, 10, 20, 21, 27, 28, 34


Course grades will be based on homework, one midterm and one final exam. Exams may have a take-home component as well as an in-class component.

Final Exam on December 13, 2007, 9:45 am
Study Guide: Use the midterm study guide. Additionally: Statements and applications of of Excision throem, Mayer-Vietoris theorem, Universal Coeff icients theorem; Definition of degree of a map of a sphere to itself, Cellular homology, Hom, cohomology groups, exact and split exact sequences; Parts or all of the following proofs: Page 129 Five lemma, Page 147 Splitting Lemma, Exercise on Page 193, counter-example to ex actness on the end; Examples of surfaces.
Homework
Chapter 0, page 18, 3, due Wednesday, 9/5/07
Chapter 0, Practice: 4-6, 9-10, 12 (path components), 14, 16, 18, 20
Chapter 0, page 18, 4 and 14, due Wednesday, 9/12/07
Compute the simpicial homology groups for the Kein bottle ( due Wednesday, 9/19/07), the 2-holed torus and the 3-holes torus ( due Monday, 9/21/07)
Prove that boundary squared is zero. Due Friday, 9/23/07
Prove that the map P constructed on Page 112 of the textbook is a chain homotopy. Due Monday, 10/01/07.
Prove in class: details of the proof of the existence of homology LES
Page 131-2: 20-22, 26, 27
Midterm Exam on October 15, 2007
Study Guide: All assigned homework, Practice Homework from Chapter 1, Pages 131-2: 1-7, 11-17, 29 (without universal covers)
Definitions of singular homology, reduced homology, relative homology groups; parts or all of the following proofs:
Boundary homomorphism squared is zero.
Propositions 2.6-7-8
A chain homotopy between chain maps induces isomorhism on homology.
Statements (i) and (ii) after Proposition 2.9 on page 111
Proof of Corollary 2.11 assuming Theorem 2.10
Corollaries 2.14, 2.15
Six steps in the proof os Theorem 2.16


Lebesgue Covering Lemma

Final homework (Due December 7, 2007)
Give an example of a space which is connected but not path-connected.
Page 19: 12, 17, 18
Page 131: 2, 11, 15, 22
Page 135: 4, 10, 20, 21, 27, 28, 34

Final Exam on December 13, 2007, 9:45 am
Study Guide: Use the midterm study guide. Additionally: Statements and applications of of Excision throem, Mayer-Vietoris theorem, Universal Coefficients theorem; Definition of degree of a map of a sphere to itself, Cellular homology, Hom, cohomology groups, exact and split exact sequences; Parts or all of the following proofs: Page 129 Five lemma, Page 147 Splitting Lemma, Exercise on Page 193, counter-example to exactness on the end; Examples of surfaces.

For details of MATH 741 from fall 2005, click here.

Back to Swatee Naik's homepage

Course Announcement
Topology is popularly known as the rubber sheet geometry, and topologists are known not to be able distinguish between a coffee cup and a doughnut, as the two define topologically equivalent (homeomorphic) spaces. Determining when two spaces are not equivalent is a difficult problem and a topologist seeks help from algebra. A homology theory assigns an abelian group to each topological space in such a way that

  • when two spaces are homeomorphic, the corresponding groups are isomorphic,
  • continuous functions between spaces induce homomorphisms between corresponding groups,
  • composition of functions is respected, and
  • identity function from a space to itself induces the indentity function from the group to itself.
    We will begin with a review of relevant topics from MATH 440 and MATH 331, study simplicial and singular homology theory, compute homology groups for various subsets of the 3-dimensional space, and study properties.

    last updated Wed Nov 21 06:22:13 PST 2007