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 cochran@math.rice.edu

Grader: email amheap@math.rice.edu  (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.
• Homework will count for 20% of the grade. It is due at the beginning of the appropriate class period. Late homework is only corrected if the grader has time and will be automatically assigned a grade of 50% of your average homework score. See hand-out for details.
• Good mathematical exposition will be counted on both exams and homework.
• collaboration is encouraged on homework porblems- see class hand-out for specifics.

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

Wednesday 1/31   Lecture  1.6, zeroth homology
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

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

Monday 2/12          Lecture:   Relative Singular Homology, Exact sequences, Eilenberg-Steenrod axioms
Read:  pp. 130132 up to 23.2 (inclusive)
pp168-169
Do; Handout for Friday 2/16

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

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
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

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)
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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.

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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
triangulation.

Wednesday 2/3:

Lecture:  Relative Homology; possibly begin singular
homology.

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

Do: Handout due Wednesday 2/10.

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Monday 2/8

Lecture: Relative Singular Homology and Homology of starlike
sets

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;
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Monday 2/15

Lecture: Eilenberg steenrod Axioms ; Chain complexes

Assignment: Skim section 26 ignoring the word simplicial
Do for Friday 2/19: p.175 # 2 and #3 using
the axioms. Hint for #3 consider
the long exact sequence of the pair
(X,A).Also
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.
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Monday 2/22

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

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
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 !!!!!!!!!!!!!!!!!!!!!!!!!

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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).

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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.

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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).

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