New, Improved!
OFFICE HOURS: 9:30-11:30am Monday, 3-4pm Wednesday
Also: please feel free to ask me questions before
class (I'll have to run after class though) or make an appointment
by email.
OFFICIAL TEXTBOOK:
"Algebraic Topology" by Allen Hatcher
This book is available for downloading
here
or you can actually buy a (cheap) bound version as well. We'll cover
Chapter 3 and ....
I'm likely to also be looking at several other books including
"Elements of Algebraic Topology" by James R. Munkres
chapters 5-8.
HOMEWORK: Usually 5 points per problem (unless otherwise noted).
Probably due every other week.
You are responsible for reading the text, preferably
before the material is covered in class. You are encouraged to work together
on all homework assignments, but write up your solutions separately and
credit your collaborators explicitly. You are also encouraged to work at
least a few of the problems by yourself.
Math Writing:
One link that has something useful to say about writing
good solutions to math problems is
here. If you find another site you like, let me know and I'll put a link
here.
Homology with coefficients:
See Hatcher p. 153-155, p.218-219, p.261-267 (or Munkres
Sections 10, 50, 51, 54, 55).
First homework, due January 25 (think of it as Jan. 23):
1. (10 pts) Compute (using cellular homology) the homology with general coefficients
G of RP^n, the connected sum of n tori and the connected sum of n copies
of RP^2.
2. Compute A tensor B and Tor(A,B) where A is the direct sum
of Z, Z/2, Z/4 and Z/6 and B is the direct sum of Z/9, Z/12 and two copies of Z.
(Why does Tor commute with direct sums?)
3. Prove that there is an exact sequence 0 -> Z/d -> Z/n -.m-> Z/n -> Z/d -> 0,
where gcd(m,n) = d.
(Hint: Show that the kernel of multiplication by m on Z/n in generated by
n/d.)
4. Problems to do on your own (not to hand in): Hatcher p.159:#40, #43.
Redo first homework, problem 1 (and possibly also 3), due Friday,
February 1.
Jason Karcher's solutions from 2006. That assignment has many problems similar to the
above problems. If you have any questions, please do ask in class
or in office hours (these solutions are
not perfect, but they are much better than the non-existent solutions
from me!)
Introduction to Cohomology
See Hatcher p. 197-204 (185-189 philosophy) or Munkres
Sections 41, 47
Second homework, due Monday, February 4:
Hatcher, p. 267, 2, 6. (1, 4 and 5 would be good to do on your
own as well.)
Third homework, due Wednesday, February 13:
1.Hatcher p. 205, #10 (read example for defining CW structure, use the given
chain comlex)
2. Calculate the cohomology with general coefficients (G) for
the spaces a) a 3-fold connected sum of RP^2's b) RP^n
c) k-fold dunce cap (RP^2 is the 2-fold dunce cap).
(I'll only grade the answers; you could
use both the universal coefficient theorem and the direct definition
from cochain complexes and then check you get the same answers.)
3. Hatcher p. 205 #11.
4.
Let $T$ be the torus and $K$ be the Klein bottle. Show that
for any map $f: T\to K$ the induced map in the second
homology group with coefficients in Z/2,
$f_*: H_2(T; Z/2) \to H_2(K; Z/2)$, is trivial.
5. Do some calculations on your own for Hom and Ext.
Drew Shulman's solutions.
If you have any questions, please do ask in class or office hours.
Cup Products: Munkres Section 48, 49; Hatcher p. 206-217.
Kunneth Theorem: Munkres Sections 56-61; Hatcher p. 218-228.
Fourth homework, due Wednesday, February 27:
1.Compute the cohomology ring of the Klein bottle using the delta complex
structure given in Hatcher, p. 102. (CORRECTED 2/21) with
Z and Z/2 coefficients. (Note, answers are needed in problem 4 below.)
2. Hatcher p. 228 #1.
3. (Munkres 49 #1) (CORRECTED, 2/20)
Let $f: S^2 \to T$ be a map from the sphere to the torus. Show that
the induced map in the second cohomology (with Z coefficients),
$f^*: H^2(T) \to H^2(S^2)$, is trivial. Conclude that
the map on the second homology (with Z coefficients) is also trivial.
What could you say about a map $g: T \to S^2$?
4.(Munkres 49 #5)
Consider the cohomology rings of the Klein bottle, K, and the
wedge of the second real projective space with a circle, RP^2 V S^1.
(NOTE: "wedge" means one point union here and is denoted by V or \vee)
a) Show that these rings are isomorphic with integer coefficients.
b) Show that these rings are not isomorphic with Z/2 coefficients.
(Note: It does not suffice to show that they have different multiplication
tables. Read about K and the connected sum of two RP^2's for another
such example ....)
5. Problems to do on your own (and maybe we should do some of these in class...): Munkres 48 #1 (Let A be a path component
of X and B be the union of the remaining path components. Assume B
is non-empty. Let c be the singular zero cochain which evaluates as 1 on each
zero simplex with image in A and evaluates as 0 on each zero simplex with
image in B. Show c is a cocycle and describe the cup product of c
with any general cohomology class b of degree n.);
Munkres 49 #2 (Show that f: X \to Y induces a trivial map
on the second cohomology with coefficients in Z/2 where a) X=S^2 and Y = K
(Klein bottle) b) X = K and Y = T, c) X = T and Y = K);
Munkres 49 #3 c) (Calculate the cohomology ring of a connected
sum of n copies of RP^2);
Hatcher p. 228 #2, #3,
Poincare Duality: Hatcher p. 230- 257, Munkres Chapter 8.
Fifth homework, due Wednesday, March 12:
1. (From Munkres 59 #5)
Compute the homology (with Z coefficients) of the
following spaces: (a) S^2 x RP^5, (b) RP^3 x RP^5, (e) S^1 x S^1 x S^3,
(f) CP^2 x CP^3.
2. (From Munkres 60 #2)
Compute the cohomology (with Z coefficients) of the
following spaces: (a) S^2 x RP^5, (b) RP^3 x RP^5, (e) S^1 x S^1 x S^3,
(f) CP^2 x CP^3. (I'll just grade the answers, you can use either
the Kunneth theorem for cohomology or the universal coefficient theorem
or both to check your answers...)
3. (From Munkres 61 #5)
Compute the cohomology ring of T x K with Z and Z/2 coefficients;
here T is the torus, K the Klein bottle.
4. Hatcher p. 229 #11.
5. Hatcher p. 229 # 6.
6. Problems to do on your own: Hatcher p. 280 #3 shows that the Kunneth
theorem for homology can not be naturally split. Hatcher p. 229 # 10
shows one needs finite generation hypotheses for the cohomology Kunneth
theorem (and comes down to a nice, but possibly tricky, algebra problem).
Hatcher p. 229 #12.
Hatcher p. 229 #8.
Compare the graded cohomology ring of the connected sum of three copies
of RP^2 (P^2 # P^2 # P^2) with the cohomology
ring of T v S^1 (the torus wedge the circle) with Z/2 coefficients.
(Either define a ring isomorphism or show one does not exist.)
Sixth homework, due Friday April 4 (or Wednesday 4/2):
1. Hatcher p. 258 #7.
2. Hatcher p. 258 #25.
3. Hatcher p. 258 #26.
4. (Munkres 68 #5 a, b, d) (Partially a "redo" of Hatcher p. 229 #6).
a) Compute the cohomology rings of H^*(CP^n) and H^*(CP^\infty).
b)Show that for any f:CP^n \to CP^n, the degree of f is a^n for some integer
a. (Degree is defined in Hatcher p. 258 #7.)
d)Show that there is no map f:CP^2n \to CP^2n of degree -1.
5. (Munkres 68 #7) Let X be a connected closed 7-manifold. Suppose
that H_7(X) = Z, H_6(X) = Z, H_5(X) = Z/2 and H_4(X) = Z + Z/3.
Give al the information you can about the cohomology groups and ring
of X (H^*(X)).
6. Problems to do on your own: Hatcher p. 258: 6, 9, p. 228: 13; Munkres
section 65: 1, 2; 66: 4; 67, 4; 68: 4,
(Munkres, 65: 1) Let M, N be compact, connected n-manifolds.
Let f: M -> N be a continuous map. Show that if M is orientable
and N is non-orientable, then the induced map in the nth homology
group with coefficients in Z/2 is trivial (f_*: H_n(M; Z/2)
\to H_n(N;Z/2) = 0)
Seventh homework, due April 16:
1. (Munkres, 22: 3b,c) Consider the two self maps f, g of the Klein bottle
displayed in Munkres. (See drawings in class, or look at p. 102 in Hatcher and
let f be a map which takes a curve "parallel" to "b" to a double curve
"parallel" to "b" and g be a map which takes a curve "parallel" to "b"
to a cuve "parallel" to "a".)
(skipping part a in Munkres) (b) Show that for any map f' homotopic to f
and for any map g' homotopic to g, f' and g' have fixed points.
(c) Find a self map h
of the Klein bottle which has no fixed points.
2. (Munkres 22: 4)
Let M be a compact smooth surface in R^n. If v(x) is a tangent
vector field to M, it is a standard theorem of differential geometry
that for some e > 0 there is a continuous map F from M x (-e, e)
to M such that as t varies F(y, t) has velocity vector v(y) at t = 0 and
F(y,0) = y. Furthermore, if the tangent vector field is non-zero at all
points in M, then there is a d such that F(y, d) is not equal to y
for 0 < |t| < d.
(a) Using these fact, show that if M has a non-zero tangent vector field,
then the Euler characteristic of M is zero.
(b) Determine which compact surfaces have non-zero tangent vector fields.
3. Hatcher p. 229: 4. (Another continuation of earlier CP^n problems,
you can assume the cohomology ring is known.)
4. (Munkres, 68: 8) Let M be a compact, orientable manifold of dimension 4n + 2.
Show that the 2n+1 homology of M cannot be isomorphic to the integers.
5.Problems to do on your own: Hatcher p. 184: 5.
Higher Homotopy Groups: Hatcher p. 337 - 348.
Eighth homework, due April 30: (Due May 7 at latest.)
1. Hatcher p. 358, #6
2. Hatcher p. 358, #8
3. updated 4/23
Assume known \pi_i(S^1) and \pi_i(S^n) for i < n.
Use U(1) = S^1 and the fibration: U(n) -> U(n+1) -> S^{2n+1}
to compute \pi_1(U(n)), \pi_2(U(n)) and \pi_3(U(n)) for n greater
than or equal to 2.
4. To do as a class: Hatcher p. 392, #33
(Should discuss in class. Seems like an interesting problem, but
I'm not sure what Hatcher means.)
FINAL: TAKE HOME FINAL, due May 7 (In my box by May 8 at 10am
at the very latest if
you want it returned graded before the prelim; in my box by
May 8 at 4pm if you want it to count for your grade.)
These problems are for you to do on your own. Do not discuss
these problems with anyone other than Brooke (a.k.a. Professor Shipley).
These problems are worth 30 points each.
Updated 5/1
1. Let M be a closed connected n-manifold such that H_k(M; Z/2) is non-zero
for some k with 0 < k < n. Prove that there can be no map f
from S^n to M which induces an isomorphism in H_n(-; Z/2).
2. Let f be a self map of S^2 x S^2 (the product of two two-spheres)
which is homotopic to the map g(x,y) = (-y, -x).
(a) Find the induced map in cohomology (f^*).
(b) Show that f must have a fixed point.
3. Consider the space X = RP^3 x RP^2.
Updated 5/1 Calculate the cohomology groups of this space with
Z and Z/2 coefficients. (Two sets of answers.)
Calculate as much as you can about the cohomology ring of this
space with Z and Z/2 coefficients. (Two sets of answers.)