MATH 734. Algebraic Topology (Spring 2008)

Meeting times: MWF, 11:00am-11:50am (MTH 0104)
Instructor: Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. His office hours are Mondays 9-10 and Fridays 1-2, or by appointment.
Homework grader: Krishna Kaipa, office in room 4410 of the Math Building.

Texts: Topology and Geometry by Glen E. Bredon, Graduate Texts in Math., vol. 139, Springer-Verlag, Corrected 3rd Printing, 1997, ISBN 978-0-387-97926-7, and Algebraic Topology by Allen Hatcher, available free on the web, also published by Cambridge University Press in a paperback edition (ISBN 978-0-521-79540-1) at $35. If you want still another reference that's not too expensive, I'd recommend A Concise Course in Algebraic Topology by J. Peter May, Chicago Lectures in Math., Univ. of Chicago Press, 1999, ISBN 978-0-226-51183-2, for $22. This book is also available on the web. It's rather terse but covers everything.

Prerequisite: MATH 403 (undergraduate-level abstract algebra). MATH 730 or equivalent is recommended, not 100% necessary if you are willing to take a few facts about the fundamental group on faith.

Catalog description: Singular homology and cohomology, cup products, Poincaré duality, Eilenberg-Steenrod axioms, Whitehead and Hurewicz theorems, universal coefficient theorem, cellular homology.

Course Description:

Basically, we will cover most of Chapters IV, V, and VI of Bredon, with some of the "starred sections" omitted. Much of this material is also in Hatcher, Ch. 2-3, with a slightly different point of view, and you might find a second presentation helpful.

Course Requirements and Grading Policy:

Homework will be assigned, collected, and graded, usually once a week. In addition, there will be a mid-semester exam and a 2-hour final exam. This course is designed to prepare students for the second half of the graduate qualifying exam in Geometry/Topology. Thus grades will have the following meaning:

40% of the grade will be based on the final exam, 20% of the grade will be based on the mid-term exam, and 40% of the grade will be based on graded homework. Homework will not ordinarily be accepted for grading after solutions have been distributed or discussed. Thus it is your responsibility to turn the homework in on time.

Course Evaluation:

Please fill out the course evaluation questionnaire between Tuesday, April 29 and Wednesday, May 14th.

Tentative Schedule

(Some details to be filled in later if necessary)

Week Material Covered (Reading Assignment) Homework and Special Notes
Week of January 28 Bredon, Ch. IV, sections 1-3. Also see Hatcher, pp. 97-110. Due Monday, Feb. 4:
  1. Let X be the non-Hausdorff T0 space with two points a and b, with {a} closed (but not {b}). Determine when a map from a simplex to X is continuous and show X is path-connected. Then show X is contractible. Show that the singular homology of X is not the same as for the (discrete) Hausdorff space Y with two points.
  2. Another homology theory, different from singular homology, but closer to Poincaré's original idea of what homology should be, is called (oriented) bordism. In this theory, n-chains are maps from smooth compact oriented n-manifolds to X, and the boundary map is retriction to the boundary. The addition operation is defined by disjoint union. (That makes the chains into a semigroup but not a group.) Thus the n-th homology group, denoted Ωn(X), consists of maps from compact oriented n-manifolds without boundary to X, modulo those that extend to a compact oriented (n+1)-manifold with boundary, with the given map as restriction to the boundary. (This is now a group!) Show that Ω0(X) is isomorphic to H0(X), the free abelian group on the path components of X. Note: By convention, a single point (which is a 0-dimensional manifold) has two different orientations, denoted + and -.
  3. (slightly harder) Continuing with the notation of #2, show that if X is path-connected, Ω1(X) is isomorphic to H1(X), the abelianized fundamental group. (Hint: A loop can be viewed as a map from S1 to X. It is null-homotopic if and only if it extends to a map from D2 to X.)
Week of February 4 Bredon, Ch. IV, sections 4-6 Due Monday, Feb. 11: Assignment 2
Week of February 11 Bredon, Ch. IV, sections 15, 7-8 Due Monday, Feb. 18: Hatcher, Ch. 2, p. 132, exercise 22, and pp. 155-156, exercises 4, 8, 12. In these problems, assume that singular homology satisfies the axioms.
Week of February 18 Bredon, Ch. IV, sections 9-11 Due Monday, Feb. 25: Bredon, p. 206, #4, 5, 6, 7, and p. 211, #4 (in fact, show it's finitely presented, i.e., is given by finitely many generators and finitely many relations).
Week of February 25 Bredon, Ch. IV, sections 12-14, 16 Due Monday, March 3:
  1. Let n > 1 and let A2, ..., An be abelian groups. Show (by induction on n) that there is a simply connected CW-complex X of dimension n+1 with Hj = Aj, j = 2, ..., n. (Hint: write each Aj as a quotient of a free abelian group by a (necessarily free) subgroup.)
  2. Show that X in #1 can be chosen to be of dimension n if and only if An is free abelian.
  3. Do exercises 21-23 in Hatcher, pp. 155-159 (all having to do with the Euler characteristic). Use #22 in the proof of #23.
  4. Let (C*, d*) be a chain complex of FREE abelian groups with Cn = 0 for n < 0. Show that if C* has all homology groups equal to 0, then C* is chain contractible. (Hint: Construct a contraction by induction on n. You will need the freeness assumption.)
Week of March 3 Bredon, Ch. IV, sections 17-19. Also see Hatcher, pp. 99-126. Due Monday, March 10: Do the problems in Hatcher, p. 133, #26 and p. 158, #33.
Week of March 10 Bredon, Ch. IV, sections 19-20. Also see Hatcher, pp. 177-184. No homework this week because of midterm exam and spring break.
Friday, March 14 Mid-term Exam on Bredon, Ch. IV, and Hatcher, Ch. 2. For help in studying, see this old exam with solutions.
Week of March 17 Spring Break, No Class  
Week of March 24 Bredon, Ch. IV, sections 21-23; Hatcher, pp. 177-184. Due Monday, March 31: Bredon, p. 250, #1-3; p. 259, #4.
Week of March 31 Bredon, Ch. V, sections 1-5 Due Monday, April 7:
  1. Bredon, problems 1 and 2 in V.2 (page 264), on de Rham cohomology.
  2. Compute the de Rham cohomology Hc,dR*(R) of R with compact supports, and show it is 0 in degree 0 and one-dimensional in degree 1. (Instead of using all differential forms, use just those supported on a compact set. All you need for this problem is first-year calculus.) Show that integration gives the isomorphism of Hc,dR1(R) with R.
Week of April 7 Bredon, Ch. V, sections 6-8. Due Monday, April 14:
  1. Assume you know that the cellular chain complex of RPn is
    ... ---> Z ---> Z ---> Z ---> ...,
    with the maps alternating between 2 and 0. Compute the cohomology of RPn in two different ways: by dualizing the complex, and by applying the universal coefficient theorem. Check that you get the same answer in both cases.
  2. (Bredon, p. 285, #6) Show that H1(X) is torsion-free, for any space X.
  3. Is the converse of the last question true? In other words, is every torsion-free abelian group of the form H1(X) for some space X?
  4. Generalize a problem on last week's assignment to show that the de Rham cohomology with compact supports of a Euclidean space is given by Hc,dRk(Rn) = 0 for k < n, R for k = n. Again, the isomorphism of the top-degree cohomology with R comes from integration over Rn.
Week of April 14 Bredon, sections 9-10 (cont'd);
Ch. VI, sections 1-2.
Due Wednesday, April 23:
  1. Do Hatcher, Ch. 3, problem 7, page 205. (In other words, show that X ---> Hom(H*(X), Z) does not satisfy all the axioms for a cohomology theory. Which axiom fails, and why?)
  2. Show that in the cohomology ring of Sn × X, cup product with the n-dimensional generator a from the first factor induces an injection Hm(X) ---> Hn+m(Sn × X). (More exactly, (a × 1) . (1 × x) = a × x and the product is non-zero unless x = 0.) This shows that the product structure on cohomology is non-trivial in this case.
  3. Deduce Bredon, Cor. VI.4.11 from the result of #2, by viewing SX as a quotient of S1 × X, and noting that the generator of H1(S1) has square zero.
Week of April 21 Bredon, Ch. VI, sections 2-5; Hatcher, pp. 206-217. Due Wednesday, April 30:
  1. This exercise provides an alternative approach to the calculation of the cup product structure on the cohomology of complex projective space. For simplicity, we stick to the case of CP2.
    1. Show that there is a map CP1 × CP1 ---> CP2 given in homogeneous coordinates by ([z0,z1], [w0,w1]) ---> [z0w1 + z1w0, z0w0, z1w1] .
    2. Show that the induced map on H2 sends the usual generator to the sum of the two standard generators, and show (from the fact that the map is generically 2-to-1) that the induced map on H4 is multiplication by 2.
    3. Use the fact that the induced map on cohomology must preserve cup products, together with your knowledge of the cup product structure for product spaces, to deduce that the square of the usual generator of H2(CP2) must be the usual generator of H4(CP2).
  2. Do the problems in Hatcher, ch. 3, pp. 228-230, #1, 7.
Week of April 28 Bredon, Ch. VI, sections 6-8. Due Wednesday, May 7:
  1. Show that the definition of orientability of a manifold Mn in Bredon, Ch. VI, section 7, is equivalent (when Mn is smooth) to triviality of the top exterior power of either the tangent or the cotangent bundle, and thus to existence of an everywhere non-vanishing differential n-form (i.e., a ``volume form''). Hint: While you can use the de Rham theorem, it is not necessary. Bredon Theorem VI.7.15 shows orientability is equivalent to existence of an atlas of charts for which all the transition functions have positive Jacobian. From the relationship between the determinant and the exterior products, show that this is equivalent to the condition above.
  2. Do Bredon, problem 6 on page 355 (the fact that the Euler characteristic of a closed odd-dimensional manifold is necessarily 0).
  3. (harder than #2) (Bredon, problem 2 on page 366) Show that if Mn is a closed orientable manifold, and if n is congruent to 2 mod 4, then the Euler characteristic of M is even. This explains why the Euler characteristic of an oriented surface is of the form 2 - 2g. Hint: Reduce to the case of M connected. Then the cup-product pairing on middle cohomology must be non-degenerate and skew-symmetric. Use the fact that any non-degenerate skew-symmetric bilinear form on a finite-dimensional real vector space can be put in the standard form represented by the matrix
    with all the blocks of size k × k for some k. (This is a non-trivial theorem in linear algebra but you may assume it.)
Weeks of May 5, May 12 Bredon, Ch. VI, sections 9-11. May 12 is day of last class.
Rescheduled to Thursday, May 15, 10:00-12:00 am, usual room (MTH 0104) Final Exam. For help in studying, here are a recent final exam and an old final exam. The latter dates to a time when we used a different textbook, but the topics covered are pretty much the same. Special Pre-Exam Office Hours: Wed., May 14, 2-4 PM
This year's exams and solutions have been uploaded to the departmental testbank.