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 !!!!
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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