12/8/16: I have posted a hint to the last homework problem pretty late
this night. I wanted to post it earlier in the evening, but remote
access to the files at the U was not working.
12/8/16: I have posted a hint to the last homework problem pretty late
this night. I wanted to post it earlier in the evening, but remote
access to the files at the U was not working.
12/6/16: My travel plans have settled, and I will travel to South
Korea during the last week of classes and have to cancel classes that
week. Homework 6 is the last homework this semester, due December
9.
12/6/16: I have made cosmetic changes today, but yesterday, I added a
hint about the universal covering of
S1∨S1&orS2.
12/3/16: I have added two problems to the last homework, #6, due
December 9. It is eight problems now - reload the page if you do not
see all the eight. I was trying to add a few problems yesterday late
night, but started falling asleep and realized that I would not do a
good job and post something insensible. :-)
11/27/16: It seems like I will likely travel to South Korea during the
last week of classes and am afraid will have to cancel classes that
week. In relation to that, I am making Homework 6 to be the last
homework this semester, due December 9. I posted it yesterday, with an
intent that it would be due December 2. I will add a few problems to
cover the material we will study before then. Thus, disregard the news
item which was posted here on 11/27/16.
11/21/16: At the very end of the proof of the Algebraic Künneth
theorem, I said that a splitting H(C⊗C') →
H(C)⊗H(C') was obtained from restricting the tensor product of
splittings s: C→Z and s': C'→Z' to Z⊗Z', which was
incorrect. It should be restricting s⊗s' to Z(C⊗C'), which
maps to Z⊗Z' with a subsequent projection to
H(C)⊗H(C'). Sorry! BTW, I am working on the homework...
11/17/16: In relation to our extended Thanksgiving Break, I am moving
my office hours on Monday, November 21, from the standard time
1:25-2:15 to 12:20-1:10.
11/10/16: Guys, I have to reschedule my office hours for tomorrow to
2:30-3:20 p.m. Or, as always, make an appointment to see me.
11/7/16: I have added a hint to Problem 1 on HW 5: Assess the
situation using a problem from the previous homework. You may assume
that tubular neighborhood of a circle fixed by r is a
cylinder.
11/5/16: Folks, we will have an extended Thanksgiving Break, from
Wednesday, November 23, through Monday, November 28. I will be out of
town during that time.
11/4/16: Added two pages of reading material (125-126) to Homework
5.
11/3/16: The last problem, #8, has been added to Homework 5.
11/2/16: Homework 5, due November 11, is posted (partially).
10/27/16: I am correcting an error in Problem 5 on Homework 4 pointed
out to me by Tikhon. The assignment there should read as: Show that at
least one of f, f2, ..., fn-1 has a fixed
point. Sorry about that.
10/24/16: I am updating Homework 4 with a hint on using cellular
homology for computing the Euler characteristic and including page
181, containing the proof of the Lefschetz fixed-point theorem, in
reading assignment.
10/21/16: Homework 4, due October 28, is posted.
10/21/16: Correction: I should have used the degree deg f of
the map f between spheres defined by a zero of a vector
field for computing the index of a vector field, rather than
(-1)deg f. See Wikipedia,
which is correct!
10/12/16: I have made a cosmetic change in Homework 3, explaining what
kind of suspension we are talking about in Problem 8. I have also
adjusted the reading assignment (pages in Section 3.3) slightly.
10/10/16: In Homework 3, Problem 1 can be done by a simple application
of the Snake Lemma, as Tikhon pointed out to me. However, I am
explicitly asking you to use diagram chasing.
10/04/16: Homework 3, due October 14, is posted. Reload the homework
page, if you do not see the new homework. Work in study groups and
come to my office hours, if you need help.
9/29/16: I have corrected a couple of typos, which many of you
probably have not noticed, in the notation of homology group in
Problem 1 on HW 2. I have changed H1 (R, K; G) to
H1 (Rn, K; Z).
9/28/16: I have made a clarification in Problem 5 on HW 2: Eric
suggested that it was not clear what the homology of a knot complement
is, because it does not automatically come with a simplicial complex
structure. I have also specified that we are talking about integral
homology.
9/28/16: I have made a correction to Problem 3 on HW 2: as Tikhon
pointed out to me, the absolute homology in the problem must be
reduced.
9/25/16: I have written up a
brushed-up proof of duality between
homology and cohomology with coefficients in a field.
9/23/16: The homework is posted. Reload the homework page, if you do
not see the new homework. Work in study groups and come to my office
hours, if you need help.
9/22/16: Our grader has graded your first homework. You have done a
great job! This also means we have got a grader. His name is June
Park. Here is how you may reach him, should you have grading-related
questions: e-mail:
junepark at math.umn.edu, office: 454 Vincent Hall, phone: (612)
624-3531.
9/15/16: In Problem 4, the hint means that you may use a result on the
homology of a short exact sequence of chain complexes, which is used
in constructing the long exact sequence of a pair: a pair (K,L), where
L is a simplicial subcomplex of K, gives a short exact sequence of
chain complexes 0 → C(L) → C(K) → C(K,L) →
0.
9/14/16: I have corrected Problem 3 on Homework 1, as two of you told
me there was an error in its wording. Sorry!
9/8/16: The first homework is posted.
9/8/16: I have started writing Class Outlines, see the course web
page. I am still working on the homework.
9/8/16: If you need to register for this course, please, send a
message to Stephanie Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register.
Last modified: (2016-12-08 23:56:55 CST)