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 !!!!

- 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.

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)

pp168-169

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)

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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

Assignment: Read: pp. 162-164

Do: Handout due Wednesday 2/10.

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Monday 2/8

Lecture: Relative Singular Homology and Homology of starlike

sets

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.

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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

(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.

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

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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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