Math 445 Algebraic Topology    Spring 2003

Professor Tim Cochran  (My home page which contains mostly research related stuff: )
Office: 416 HB, office hours  W 2-3, Thur 1-2    and by appointment ;
(713 348 5265) or email

Grader: email  (Aaron Heap HB 050  x2841)

Hey you guys- come to office hours- look how friendly I look !!!!


Text:  Elements Of Algebraic Topology,  James Munkres
Topics: Introduction to algebraic methods in topology . Simplicial complexes, Homology theory, Cohomology theory , Poincare Duality.

  • Grading: There will be a final exam and one short mid-term exam, both take-home.
  • Lectures and Homework Assignments:

    Wed. 1/17  Lecture: introduction; simplices
                       Read: pp.2-5 and pp 7-10 stressing basic definitions

    Friday 1/19    Lecture section 1.2   section 1.3
                           Read: pp 15-17  and review pages 20-25; review quotient groups
                           Do: #1c and #2 on p. 19-20 and hand in monday 1/22

    Monday 1/22   Lecture 1.5
                            Read: section 1.5 but omit Munkres definition of a p-chain and use the one given in class.
                            Do: p.33  #2,4,5 extra credit #6b and hand in Friday 1/26

    Monday 1/29        lecture: start 1.6
                                 read: 1.6 carefully

    Wednesday 1/31   Lecture  1.6, zeroth homology
                                 Read: 1.6,  1.7
                                  Do: start HW due Friday

    Friday  2/2          continue

    Monday 2/5      Lecture:  Relative Homology  , start singular homology
                            Read:  47-49  and first sentence of p.50 especially study examples 1,2,3

    Wednesday 2/7    Lecture: singular homology
                               Read: 165-168

    Friday 2/9       Lecture: Homology of star-like sets
                           read: pp165-168

    Monday 2/12          Lecture:   Relative Singular Homology, Exact sequences, Eilenberg-Steenrod axioms
                                   Read:  pp. 130132 up to 23.2 (inclusive)
                                               section 26 ignoring the word simplicial (see also pp.168-170).
                                     Do; Handout for Friday 2/16

    Wednesday 2/14    Lecture: finish ES Axioms, chain complexes, start zig-zag lemma
                                 Read:  71-72

    It is time to review the following terms: homotopy between maps, homotopy equivalence, retract, deformation retract

    Friday 2/16   Lecture:proof of Zig-Zag lemma and proof of Exactness Lemma
                         Read: section 24
                         Do handout which is due next wednesday 2/21

    ----------------STOP : What you see below this line is from last year. Don't read it.----------------------------------------------------------------------------------------------

    Friday 1/28/2000

                             Lecture: finish section 5 and start section 6
                             Assignment: Read section 5 and work on problems which are due on Monday
                                                  Review algebra topic : Quotient groups

    Monday 1/31/2000

                             Lecture: start section 6
                             Assignment: Do. p. 40 # 2 (this is tedious- don't include all details- this is warm-up for #4) DO NOT HAND THIS IN
                                                Do. p. 40 # 4,6,7,8  (actually #6 is easier- you could start with it)  Hand these in Monday 2/7  , BUT START IT BY 2/2 !!!!!!!!!!!!!!!!!!!!!!

    Wednesday 2/2/2000

                             Lecture: section 6;
                             Assignment: Read section 7 and begin work on homework from page 40

    Friday 2/4/2000

                            Lecture:Finish section 6;  Homology in dimension zero.

    Monday 2/7/2000    **********OFFICE HOUR #PM CANCELLED TODAY***
                                                         REPLACED BY TUESDAY 12-2pm

                            Lecture: Reduced homology, Relative Homology
                            Assignment:  Read pp. 47-49 and the first sentence of page 50. Especially study examples 1,2,3.

    Wednesday 2/9/2000

                            Lecture: Singular Homology
                            Assignment: Read pp. 162-164, possibly 165-167; Do Handout  for next wednesday 2/16

    Friday 2/11/2000

                           Lecture: Relative Singular Homology; Homology of star-like sets
                           Assignment:  Read pp. 165-169

    Monday 2/14/00

                           Lecture:  same as 2/11 (continued)

    Wednesday 2/16/00

                          Lecture:  Exact Sequences
                          Assignement:  Read : pp. 13-132 up to and including 23.2;  Do handout   due 2/23

    Friday 2/18/00

                         Lecture:  Eilenberg-Steenrod Axioms
                         Read: section 26 ignoring the word simplicial, also see pp. 168-170

    Monday 2/21/00
                         Lecture: Finish Axioms and applications
                          Assignment:  See Theorem 30.8 p.174; good students start p. 175 #2,3

    Wednesday 2/23/00
                          Lecture: Chain Complexes; Zig-Zag Lemma; proof of Exactness Axiom
                          Assignment: Do handout due 3/1

    Friday 2/25/00
                        Lecture: Finish Zig-Zag Lemma; Start proof of Homotopy Axiom
                        Assignment: Read pp.64-66  pp.72-73 (motivation for chain homotopy);   pp. 170-175 (actual proof)

    STOP    STOP     STOP

    IGNORE ALL BELOW THIS LINE_ THIS IS FROM 1999 !!!!!!!!!!!!!!!

    Wednesday 1/20 :

        Lecture: Introduction and Motivation; Quick Review of 1.2

        Assignment: Review pp.2-5, 7-10 stressing basic definitions and  Review pp.20-25 especially 20-21.

                    DO: p.25  # 1 and 2 (due 1/27 in class)

    Friday 1/22 :

        Lecture:  1.5

        Assignment: Read pp. 15-17.5 (here I mean half-way down page 18). Read section 1.5 but omit the definition of p-chain on page 27 and use the one from class.

      DO: p.19 1b , 2
          p.33 #1-5 (one thru 5) but do not hand in #1. Extra credit #6b. All due 1/27 in class.

     OPTIONAL: Read pp. 147-148 of Massey (book from 444)
    Monday 1/25:

        Lecture: 1.5

    Wednesday 1/27:

        Lecture: 1.3 and start 1.6

        Assignment: Read 1.6. Do p.40 #2,3 but do not hand in-they
    are preparation for #4.

            Do: p.40 #4,6,7,8 and hand in on Wednesday 2/3

    Friday 1/29:

       Lecture: 1.6

       Assignment: Read 1.7 if we get to it in class. Do
                    p.43 #1 but do not hand in.

     Monday 2/1:

       Lecture: Finsih 1.6, zero-th homology and reduced homology

       Assignment: Read 1.7, work on problems due Wednesday 2/3.
                   Try to use the techniques of 1.6. Don't worry
                    too much about the details of the

     Wednesday 2/3:

       Lecture:  Relative Homology; possibly begin singular

       Assignment: Read: pp.47-49 and first sentence on p. 50,
                   especially study examples 1,2,3.

     Friday 2/5:

       Lecture: Singular Homology; possibly 165-167

       Assignment: Read: pp. 162-164
                   Do: Handout due Wednesday 2/10.


    Monday 2/8

      Lecture: Relative Singular Homology and Homology of starlike

      Assignement: Read pp.165-169
      Do: Prove line 4 p. 169 for Friday 2/12  !!!

    Wednesday 2/10

      Lecture: Finish Homology of Starlike sets
               Exact sequences

      Assignment: Read: pp. 130-132 (up to and including 23.2)
                  Do: Handout for Wed. 2/17

    Friday 2/12

      Lecture:  Eilenberg-Steenrod Axioms

      Assignment: Read section 26 ignoring the word simplicial;
                   see also pp.168-170.

    Monday 2/15

         Lecture: Eilenberg steenrod Axioms ; Chain complexes

         Assignment: Skim section 26 ignoring the word simplicial
                     Read pp. 71-72
                     Do for Friday 2/19: p.175 # 2 and #3 using
                                  the axioms. Hint for #3 consider
                     the long exact sequence of the pair
                      stuff about exact sequences such as on
                       page 130.

    Wednesday 2/17:

        Lecture: Zig-Zag lemma; proof of Exactness Axiom

        Assignment:  section 24 Handout for Wed. 2/24

    Friday 2/19:

        Lecture: Begin proof of Homotopy axiom

        Assignemt: Read: pp64-66, pp.72-73 on motivation for
                    the proof and definition of chain homotopy.
                     Read :pp 170-175 for actual proof.

    Note: There was a mistake on assigments: Problem #1 p.175
    is the same as problem #1 p. 141. You need only do one of them!! Problem #3 p.175 was assigned twice with two due
    dates. The earlier one applies. But problem #1 can be turned
    in 2/24.

     Monday 2/22

      Lecture: chain homotopy; proof of excision Axiom except Theorem 31.5; example of use of Excision Axiom.

      Assignment: Read: p.66, pp 72-72.5
                  Read: pp.179-182 skipping proof of 31.5 for now.
                   Look back to page 50 to see how easy it is to
                    prove Excision for simplicial homology.

    Extra problem due Wednesday 2/24: p.141 #3a  Extra credit:
    which hypotheses are not used ? The 5-Lemma is used often.
    Use diagram chaising to see the proof.

    Wednesday 2/24:

      Lecture: Mayer - Vietoris Theorem
      Assignment: Read section 33
                   Do handout for friday 3/5. This is a long and
                   important and difficult assignment.

    Friday 2/26

      Lecture : Applications of Mayer-Vietoris theorem

      Assignemnt: Skim sectio 36 to see a few more aplications
                 Begin Homework !!!!!!!!!!!!!!!!!!!!!!!!!


    Monday 3/1:

      Lecture: Proof of theorem 31.5, R^n is not homeomorphic to R^m

      Assignment: Read: section 31;   pp. 85-86

    Wednesday 3/3:

      Lecture:  Cell complexes; effect on homology of adding a cell

      Assignment: Review quotient spaces a little if you like ( see sections 20 and 27 of our text)

                  Do: Read handout on attaching cells by next week.

                       and do  handout of problems due wednesday 3/10

     Friday 3/5:

       Lecture: Euler characteristic;  CW complexes

       Assignment: Optional reading: Greenberg and Harper sections 19 and 20 (on reserve at library)

                   Read: Massey on CW complexes (on reserve at library for those who didn't take 444) (our
                        text has a different but equivalent definition of CW complexes).


    Monday 3/15:

      Lecture: Euler Charachterisitic, CW-complexes and examples

      Assignment: Midterm due Monday 3/22 in class; for CW complexes see handout on adding cells and see pp. 231-232 of
                  our Munkres text.

    Wednesday 3/17:

      Lecture: RP(n) and CP(n)

      Assignment: Read pp. 233-234 and also see  handout on adding cells ; Do handout for Wednesday 3/24

    Friday 3/19:

      Lecture: Cellular Homology

      Assignment: Read 222-224 using lecture notes more than text as regards Cellular homology.

     Monday 3/22:

       Lecture: Homology of CP(n) and RP(n)

       Assignement: work on problems due 3/24

    Wednesday 3/24:

      Lecture: more on cellular homology

       Assignment: Skim munkres chapter on cellular chain complex

    Friday 3/26:

       Lecture: Categories and Functors; the HOM functor  ;  Cohomology of a chain complex (if time)

       Assignemnt: Read pp. 155-158  (skip "natural transformation- this is a very fancified way of saying various
    things are "natural" which I have given a metadefinition for and will again for your edification)
                   Read section 41  (actually we will not discuss theorem 41.2 for a little while but you could look at it now if you want)
                   Read: 262-264 if we get to cohomology in class.

                    Do: Handout for wednesday 3/31  (you can do several of these without Friday's lecture so DO IT NOW).