# Introduction to Algebraic Topology

## 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.
 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).
 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).

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).
 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!
 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).

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).
 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).
 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).

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).
 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.
 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).

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).
 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).
 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).

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).
 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).
 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.

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).
 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).

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).
 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).
 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).
 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.

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).