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Homework assignments
Please note the change in the homework
due dates
Homework assignments are to be handed in
weekly. In general, assignments for Wednesday and Friday on a given week
as well as the one for next Monday are to be submitted in class
next Friday.
| Week of May 23 – May 27, 2005 |
Friday, May 27:
-
Study:
Chapter 8, §37.
-
Do written assignment (due Wednesday, June 1,
at the end of the lecture):
Problems 37:O, 37:P (Hint: compare to 37.1), and 37.11 (Hint: find such a
cellular space Y that both spaces in question can be obtained from it
by elementary collapses; Y is constructed by attaching 3 cells to
X: two n-dimensional (via φ and ψ) and one
(n+1)-dimensional. Which one??) (page 203). Problem for extra
credit (and lots of glory): 37.12.
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Wednesday, May 25:
-
Study:
Chapter 8, §37.
-
Do written assignment (due Wednesday, June 1,
at the end of the lecture):
Problems 37:I, 37:J, 37:M (page 202).
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Monday, May 23:
-
Study:
Chapter 8, §37.
-
Do written assignment (due Friday, May 27,
at the end of the lecture):
Problems 37.F (compare with 37.E), 37.1, 37.G, and 37.4 (see §20°14,
page 128, for the definition of spheres with handles and crosscups) (pages 199,
200).
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| Week of May 16 – May 20, 2005 |
Friday, May 20:
-
Study:
Chapter 7, §36.
-
Do written assignment (due Friday, May 27,
at the end of the lecture):
Problems 36:D (Hint: note the colon in the problem number) (page 193), 36:I
(read the definition of covering automorphisms on page 194), 36:J, and 36:11
(Hint: take a bouquet of two circles as the base space) (page 195).
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Wednesday, May 18:
-
Study:
Chapter 7, §36.
-
Do written assignment (due Friday, May 27,
at the end of the lecture):
Problem 35.19 (page 189) and the following one: Let B be a
path-connected space. Which covering of B is subordinate to any other
covering of B? Which covering of B has every other covering of
B as its subordinate? Justify your answer!
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Monday, May 16:
-
Study:
Chapter 7, §35.
-
Do written assignment (due Friday, May 20,
at the end of the lecture):
Problems 35.15, 35.17, 35.20.(h,k,l), 35.21 (Hint: what is the fundamental
group of
R3\{core of the Möbius band}? Which
element in this fundamental group does the boundary of the Möbius band
represent?) (pages 189, 190).
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| Week of May 9 – May 13, 2005 |
Thursday, May 12:
-
Study:
Chapter 7, §35.
-
Do written assignment (due Friday, May 20,
at the end of the lecture):
Problems 35.9 (Hint: find a simple subspace of R^3 such that both spaces
in questions can be embedded into it as strong deformation retracts), 35.10,
and 35.11 (Hint: use the idea from 35.12 (page 188).
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Wednesday, May 11:
-
Study:
Chapter 7, §35.
-
Do written assignment (due Friday, May 20,
at the end of the lecture):
Problems 35.6 (Hint: think of a torus as being made out of a square), 35.8
(consider capical letters only) (page 188).
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Monday, May 9:
-
Study:
Chapter 7, §34.
-
Do written assignment (due Friday, May 13,
at the end of the lecture):
Problems 34.8, 34.13, 34.14 (Hint: prove by contradiction, if f is a map
without a fixed point, consider maps
 and
 What can you say about them?) and 34.15
(pages 185, 186).
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| Week of May 2 – May 6, 2005 |
Friday, May 6:
-
Study:
Chapter 7, §33 and §34.
-
Do written assignment (due Friday, May 13,
at the end of the lecture):
Problems 33:B and 33:C (don't forget to justify your answer) (page 182)
and problem 34.6 (compare to 34.3 and 34.4) (page 184).
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Wednesday, May 4:
-
Study:
Chapter 7, §33.
-
Do written assignment (due Friday, May 13,
at the end of the lecture):
Problems 33.7 (Hint: prove by contradiction, use 33.K for n=2) and 33.8 (Hint:
solve 33.8.1 first by using 33.K for n=2) (page 181). Problem for extra
credit: 33.9.
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Monday, May 2:
-
Study:
Chapter 7, §33.
-
Do written assignment
(due Friday, May 6,
at the end of the lecture):
Problems 33.2 and 33.3 for the space
S1×S1\(1,1), a punctured torus, only (Hint: use the idea from 33.3.1; recall that we
know a single example of a space with an infinite non-cyclic fundamental group
so far) (page 180).
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| Week of April 25 – April 29, 2005 |
Friday, April 29:
-
Study:
Chapter 6, §32.
-
Do written assignment
(due Friday, May 6,
at the end of the lecture):
Problems 32.12 (page 178) and the following one: construct the universal
covering and compute the fundamental group of the bouquet of two
circles S1 and a sphere S2 (three spaces to
form a bouquet of in total).
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Wednesday, April 27:
-
Study:
Chapter 6, §32.
-
Do written assignment
(due Friday, May 6,
at the end of the lecture):
Problems 32.1 (page 174) and 32.10 (page 178).
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Monday, April 25:
-
Study:
Chapter 6, §31.
-
Do written assignment
(due Friday, May 6,
at the end of the lecture):
Problems 31.4 (Hint: how many sheets can such covering have?) and 31.5 (pages
172, 173).
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| Week of April 18 – April 22, 2005 |
Friday, April 22:
-
Study:
Chapter 6, §30 and §31.
-
Do written assignment (due Wednesday, April 27,
at the end of the lecture):
Problems 30.8 (this one can be tricky, think carefully), 30.14 (page 170) and
problem 31.3 (page 172).
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Wednesday, April 20:
-
Study:
Chapter 3, §21 and Chapter 6, §30.
-
Do written assignment (due Wednesday, April 27,
at the end of the lecture):
Problems 21.1 (Please find geometric
description of both homeomorphisms in question; note that the
problem is about all the lines on the plane, not only those that pass
through the origin), 21.2 (page 131) and problem 30.10 (page 170).
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Monday, April 18:
-
Study:
Chapter 6, §30.
-
Do written assignment (due Wednesday, April 27,
at the end of the lecture):
Problems 30.5 and 30.18, 30.6 and 30.19, 30.7, 30.24 (Please note the obvious
typo: the map in question should be
q°p:X→Z;
Where do you use the fact that q has finitely many sheets?) (pages 170, 171).
Don't forget to justify your answers! Problem for extra credit: 30.23.
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| Week of April 11 – April 15, 2005 |
Friday, April 15:
-
Study:
Chapter 5, §29 and Chapter 6, §30.
-
Do written assignment (due Wednesday, April 20,
at the end of the lecture):
Problem 29.I (page 167) and problems 30.E, 30.1, 30.2, 30.3 (pages 169, 170).
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Wednesday, April 13:
-
Study:
Chapter 5, §28 and §29.
-
Do written assignment (due Wednesday, April 20,
at the end of the lecture):
Problems 29.1, 29.5, 29.4, and 29.3 (Hint: read comments after the problem
29.5) (pages 166, 167). Problem for extra credit (and glory): 28.2
(Hint: solve for r=2 first, and then use induction) (page 162).
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| Week of April 4 – April 8, 2005 |
Friday, April 8:
-
Study:
Chapter 5, §28.
-
Do written assignment (due Wednesday, April 13,
at the end of the lecture):
Problems 28.11 (don't forget to justify your answer), 28.12, 28.13 (pages 164,
165).
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Thursday, April 7:
-
Study:
Chapter 5, §28.
-
Do written assignment (due Wednesday, April 13,
at the end of the lecture):
Problems 28.8, 28.H (Hint: use 26.K), 28.10 (page 163).
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Wednesday, April 6:
-
Study:
Chapter 5, §27.
-
Do written assignment (due Wednesday, April 13,
at the end of the lecture):
Problems 27.1, 27.2 (recall that an indiscrete topological space is the one
where only the empty set and the whole space are open), 27.3 (pages 158, 159).
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Monday, April 4:
-
Study:
Chapter 5, §26.
-
Do written assignment (due Friday, April 8,
at the end of the lecture):
Problem 26.4 (page 155) as well as
two problems from this file.
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| Week of March 30 – April 1, 2005 |
Friday, April 1:
-
Study:
Chapter 5, §26.
-
Do written assignment (due Friday, April 8,
at the end of the lecture):
Problems 26.7, 26.8, 26.10, 26.13 (Hint: consider the line segment [f(x),-x]
in Rn+1; how does it help?) (page 155).
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