MATH 7512: Topology II Homework

MATH 7512 Topology II

Spring 2005



Munkres page 493 #3
An additional problem:
  1. Let M be an n-manifold. Show that M is semilocally simply connected, and that M has a universal cover which is also an n-manifold.

Munkres page 487 turn in #1(a)Friday, April 22
Munkres page 492 #1 - 3 turn in #2(a)Friday, April 22
 

Munkres page 483
 
#1 - 7
 
turn in
 
#3, 4
 
Friday, April 15
(these may be turned in on April 22)

Munkres page 457 #3, 4 turn in #3Friday, April 8
An additional problem for Friday, April 8:
  1. Let K be a knot in R3, and view S3 as the 1-point compactification of R3 (so K is in S3).
    Show that the fundamental groups of R3 \ K and S3 \ K are isomorphic.

Munkres page 421 #2, 3 turn in #2(a)Friday, March 18
Munkres page 425 #1, 3 turn in #1Friday, March 18
Munkres page 438 #2, 3 turn in #3Friday, March 18
Two additional problems for Friday, March 18:
  1. Let A be the union of n distinct lines through the origin in R3.
    Determine the fundamental group of R3 - A.
  2. Let G be the fundamental group of the "double torus" T#T (see Figure 60.2 on Munkres page 375).
    Find a presentation for G, and determine the structure of G/[G,G].

Munkres page 359 #2, 3, 4 turn in #2, 4(a)Friday, March 4
Munkres page 375 #2, 4, 5 turn in #2Friday, March 4
Munkres page 411 #1 - 6 turn in #6Friday, March 4

Munkres page 366 turn in #5, 9Friday, February 18
Munkres page 375 #1, 2 turn in #1Friday, February 18

Munkres page 341 #1, 3 - 6 turn in #1, 3, 4, 6(b)Friday, February 11
Munkres page 347 turn in #3, 6, 8Friday, February 11
Munkres page 353 turn in #2Friday, February 11
Munkres page 370 turn in #3Friday, February 11

Munkres page 330 #1 - 3turn in #1, 3(a), 3(b)Thursday, January 27
Munkres page 334 #1 - 5turn in #1(b), 3, 4Thursday, January 27

Dan Cohen                           Spring 2005
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