Spring-2020. Math 215B (cnn 24023): Algebraic Topology

Instructor: Alexander Givental
Lectures: TuTh 3:30-5 in 736 Evans
Text: A. Fomenko, D. Fuchs, Homotopical Topology (Graduate Texts in Mathematics), 2nd ed., 2016, available to our library patrons in electronic format
Homework: due weekly on Th in class
Office hours: Tue 5-8 pm in 701 Evans
Grading policy: Here is one I tried successfully in some courses and intend to use this time. The starting point is: 50% homework + 50% take-home final (most likely during the RRR week). Next: each individual weekly hw score which is lo\ wer (percentage-wise) than your score on the final will be dropped - together with its weight. E.g.: if half of your hw is below your final score and half above, then your total score is composed 2/3 from the final score and 1/3 from the remaining hw. Thus, there are many reasons why you want to do hw (as well as \ many other exercises, not assigned as hw), and do it well, yet a particular hw score can only improve your overall performance, but can never hurt your ultimate result compared to the final exam. Besides, I don't have a preconceived distribution of As and Bs, and would be happy to give everyone an A, should every\ one demonstrate good knowledge of the subject.
Course outline: In 215A, we were following the book "Homotopical topology" by Fomenko and Fuchs to cover the essence of Chapters I and II: homotopy theory, followed by (co)homology theory up to intersection theory on manifolds, including classification of principal and vector bundles over cellular bases, and a primer of the theory of characteristic classes. In 215 B, we'll begin with obstruction theory ('Lecture 18' in the book) to lay down more solid foundations for the theory of characteristic classes, then proceed to Chapter III on spectral sequences, perhaps learn something from Chapter IV on cohomological operations, then skip Chapter V on Adams' spectral sequence, and then possibly spend some time on complex K-theory and complex cobordisms, or maybe deviate from the book toward equivariant cohomology and localization formulas, or will do both if time permits.

HW1, due Th, Jan. 30: Read (a dence subset of) Lectures 18 and 19. Solve:
1. Exercise 1 from Lecture 18.
2. Show that the Chern classes c_m of a complex vector bundle and its complex conjugate (defined near Exercise 1 in Lecture 19) differ by the sign (-1)^m, and derive from this that for the complexification of a real vector bundle, 2 c_m =0 whenever m is odd.
3. Prove that Chern, Pontryagin, and Stiefel-Whitney classes are stable: c_m(V\oplus \C)=c_m(V), p_m(W\oplus \R)=p_m(W), w_m(W\oplus \R)=w_m(W), while the Euler class is not.
4. Use the classification theory of vector bundles and the Schubert cell structure of grassmannians in order to prove that a complex vector bundle over a 2n+1-dimensional cell space is stably equivalent to an n-dimensional bundle, and use this to express Chern classes as the Euler classes of suitable vector bundles over suitable skeletons of the base.

HW2, due Th, Feb.6: Read Lecture 20 (.1-.4), Lecture 21, Lecture 22.1. Solve:
1. Prove that the 1st Chern class of the tensor product of complex line bundles is the sum of the 1st Chern classes of the factors: c_1(L \otimes L')=c_1(L)+c_1(L').
2. The group Sp_1 of unit quaternions is identified with SU_2 which in its turn is a double cover of SO_3=SU_2/(\pm 1). Consequently the circle T^1 of diagonal matrices in SU_2 double-covers the "maximal torus" SO_2 \subset SO_3. So, we have BT mapped to BSU_2 mapped to BSO_3 and BT mapped to BSO_2 mapped to BSU_2. Show that H^*(BSp_1)=\Z [u] where \deg u=4, and analyze how u is related by the above maps with the 1st Pontragin class p_1 \in H^4(BSO_3), and with the respective generators x, y of H^*(BT)=\Z[x] and H^*(BSO_2)=\Z[y].

Let \H^n denote the space of n-columns of quaternions made a right quaternionic vector space by termwise colum addition and termwise multiplication of the columns by quaternionic scalars on the right . The space can be equipped with the quaternion-valued "Hermitian" form (a,b)=a_1*b_1+...+a_n*b_n (where * denotes quaternionic conjugation), which is \H-linear with respect to b and \H-anti-linear with respect to a (check this!) The group Sp_n of automorphisms of this form is a maximal compact subgroup in the group GL_n(\H) of all nxn-martices with quaternionic entries (acting on \H^n the usual way - via multiplication of a column by a matrix on the left ).

In the group Sp_n consider the subgroup Sp_1 x ... x Sp_1 of diagonal matrices of unit quaternions, and inside it consider the subgroup T x ... x T (where the circle T in Sp_1=SU_2 is described in the previous exercise).

3. Prove that the quaternionic grassmannian \H G(\infty, n) is BSp_n.
4. Prove that the inclusion Sp_1^n into Sp_n induces the inclusion of H^*(BSp_n) as a subring of symmetric polynomials into the ring \Z[u_1,...,u_n] (whose generators u \in H^4(BSp_1) were introduced in the previous exercise), and that the inclusion of T^n into Sp_n identifies H*(Sp_n) with the subring in H^*(BT^n)=\Z[x_1,...,x_n] consisting of symmetric functions of the squares of x_i.

HW3, due Th, Ferb. 13: Read Lecture 23 (1-3,5) and Lecture 24. Solve:
1. Prove that the quaternionic Hopf line bundle over \H P^1 can be considered as a 2-dimensional complex vector bundle, but is not equivalent to the complexification of any real 2-dimensional bundle.
2. Cohomological version of the spectral sequence of a fibration (X,B,F,\pi): Describe the needed filtration on the cochain groups C^*(X) and explain why the term E_1^{p,q} consists of cellular p-cochains of the base with coefficients in cohomology H^q(F) of the fiber.
3. Prove part (1) of Proposition on p. 338. (I thought I gave a proof in class, but based on the reaction of the class, it would make sense to see your explanation.)
4. For the fibration of the point over K(\pi,n) with the fiber K(\pi,n-1) (where \pi is an abelian group), examine the homological spectral sequence with coefficients in \Z and the cohomological one with coefficients in \pi. Recall the basic definitions (perhaps, of homotopy and homology sequences of pairs and Hurewicz homomorphisms) in order to explain why the transgression from E^n_{n,0} to E^n_{0,n-1} is the identity map from \pi to \pi. Show that the cohomological transgression from H^{n-1}(K(\pi,n-1);\pi) to H^n(K(\pi,n);\pi) maps the fundamental class to the fundamental class.

HW4, due Th, Feb. 20: Read: Lectures 25, 26, 27.1. Solve:
1. (to justify the use of spectral sequences for Serre fibrations)
For a Serre fibration \pi: X \to D^p over a disc, show that H^{p+q}(X, \pi^{-1} dD; G) = H^q(F; G), where F is the fiber over (say) the center of the disc.
2. Prove that in any principal U_n-bundle, the generators x_1,x_3,...,x_{2n-1} of the cohomology algebra H^*(U_n) are transgressive.
3. Compute the cohomoilogy ring of the space of non-zero vectors in the total space of the tautological (universal) complex vector bundle over \C G(\infty, n). (Use spectral sequences, and then check your answer by identifying the space with a more familiar one - or the other way around.)
4. Perhaps the initial problem 4 was too complicated. So, I am changing it to the following one: Compute H^6(K(\Z,3);\Z) and identify in it the square e^2 of the fundamental class e from H^3(K(\Z,3);\Z).

HW5, due Th, Feb.27: I plan to give a review of sheaves and Cech cohomology. The reading for this week is from "Principles of Algebraic Geometry" by Griffiths-Harris (available electronically for our library partrons), pages: 35-45 (on sheves and cohomology) and 463-468 (on Leray spectral sequences). Solve:
1. Computations of (co)homology of killing spaces X|_n using spectral sequences of fibrations are applied in the book for n>2 and simply connected X. Try to use it for n=2 and non-simply connected X and find out why this doesn't work the same way as for n>2. In particular, explain how the difficulties manifest in the counter-example X=S^1 \wedge S^2 following Theorem on p. 369.
2. Read A. Borel's theorem on p. 402, and apply it to prove that H^*(BSO_n; \Z_2) = \Z_2[w_2,...,w_n], a polynomial algebra with generators of degrees 2,...,n. (We alredy know this from the theory of Stiefel-Whitney classes, but I am asking you to derive it using Borel's approach.)
3. Prove that H^*(V(2n+1,2m;\Q)=H^*(S^{4n-1}xS^{4n-5}x...xS^{4n-4m+3};\Q) and that the generators x_{4n-1},x_{4n-5},... of these exterior algebras map to the respective generators in H^*(SO_{2n+1};\Q). Hint: Consider the map between spectral sequences of the bundles induced by the map SO_{2n+1}=V(2n+1,2n) \to V(2n+1,2m) of the total spaces fibered over the same base V(2n+1,2), and apply induction. Remark: This exercise implies universal transgressivity of the generators x_3,x_7,...x_{4n-1}.
4. Describe a complex line bundle over a manifold M, using an appropriate open cover { U_a }, by transition functions \exp 2\pi i f_{ab} between trivializations over U_a and U_b (defined on their intersection), and show that f_{bc}-f_{ac}+f_{ab} is a 2-cocycle in the Cech complex C^*(M;\Z), where \Z is the constant sheaf on M. Remark: In fact this cocyle represents the 1st Chern class of the line bundle.

HW6, due Th, Mar. 5: Read Lectures 28, 29. Solve:
1. Prove the algebraic "Poincare Lemma": For k>0, a polynomial (hence holomorphic) closed differential k-form in \C^n is the differential of a polynomial differential (k-1)-form.
2. Let A be a closed 1-form on a manifold M. Consider the complex of sheaves (\Omega^p(M), D) of differential forms on M with the differential D:=d-A \wedge, and argue that it is an acyclic resolution of the locally constant sheaf whose local sections can be described as functions of the form C\exp (\int A). Explain how to compute the cohomology with coefficients in this sheaf using (a) De Rham-like complex, and (b) CW-structure (or triangulation) on M.
3. Compute cohomology of \C - {z_1,...,z_n} with coefficients in the locally constant sheaf whose sections are single-valued branches of the finction (z-z_1)^{a_1}...(z-z_n)^{a_n}, where a_1,...,a_n are given complex numbers (not all integer), and z is the coordinate on the complex line \C.
4. Let (z^n-w)^{\a} be sections of a locally constant sheaf on \C^2-{(z,w):z^n=w}, where \a is a given non-integer complex number. For the projection (z,w) \mapsto w (\in \C-0), compute the action of the fundamental group of the base on the cohomology of the fiber with coefficients in this sheaf.

HW7, due Th, Mar. 12: Read 30.1,30.3-5, 31.2. Solve:
1. The identity map \Omega X \to \Omega X induces \Sigma \Omega X \to X and hence homomorphisms H^q(X;G)\to H^q(\Sigma \Omega X; G) \to H^{q-1}(\Omega X; G) where the 2nd arrow is inverse to suspension. Show that the (possibly multi-valued) inverse homomorphisms are the transgressions in the G-coefficient spectral sequence of the fibration (pt) \to X with the fiber \Omega X.
2. Describe reduced \Z_2-cohomology algebras of X\wedge Y and X \smash Y in terms of reduced \Z_2-cohomology algebras of X and Y.
3. Prove that the Steenrod squares are additive: Sq^r(a+b)=Sq^r(a) + Sq^r(b). Hint: Perhaps you'll need to recall the construction with Eilenberg-MacLane spaces that induces addition in cohomology.
4. Prove the formula for Steenrod squares Sq^k w_m of Stiefel-Whitney classes - or reconstruct the details of the outline given in section 31.2.

HW8, due Th, Mar.19: Read: 44.1. Solve:
1. Prove Wu's formula from 31.2 (expressing Stiefel-Whitney classes of a closed manifold in terms of Steenrod squares and other homtopy invariant data).
2. Identify the Thom space of the tautological bundle over G(N,n) with G(N+1,n)/G(N,n-1) (where the standard embedding of the grassmannians is assumed), and generalize this to complex grassmannians.
3. A (not very complicated) theorem says that given a finite regular G-covering X'\to X (not necessarilt connected), for a field \F of characteristic p not dividing |G| (including p=0), H^*(X;\F)=H^*(X';\F)^G, the G-invariant part of the cohomology of the total space X'. Rather than proving the theorem in general, consider the special case \F=\Z_p, G=\Z_2 using the following general plan. The covering can be induced by a map X \to BG=K(G,1) (from the universal cover of BG); convert it to the (homologically non-simple) fibration with the fiber X', and apply the (generalized) Leray spectral sequence with E_2 = H^*(K(G,1); "H^*(X';\F)"), where "H^*(X';\F)" is a locally constant sheaf.
4. Given a smooth map f: X^m \to Y^n of oriented closed manifolds, consider an oriented vector bundle E\to Y over Y of dimension > 2m-n, perturb f into an embedding of X into E, and describe the direct image f_!: H^*(X)\to H^{*-m+n} \to H^*(Y) in terms of suitable Thom isomorphisms. Show that the resulting homomorphism does not depend on the choices of the bundle and perturbation.

HW9, due Th, April 2: Read 38.5, 42.1, 44.3.B, 44.A-B. Solve:
1. Use Thom's transversality theorem to show that on a compact manifold, a generic function of class C^k with k>1 is Morse (i.e. has only non-degenerate critical points). Also, show that a generic vector field of class C^k with k>0 has only non-degenerate zeros. Does the 1st statement follow from the 2nd applied to the gradient vector field of the function?
2. For a manifold X, give a geometric description of the multiplication in the ring \Omega_O^*(X) of unoriented bordisms in "Poincare-dual terms" (i.e. a description that in the case X=point would coincide with the Cartesian product of manifolds).
3. Verify the table on p. 593 (I think there are errors).
4. Prove the formula (at the bottom of page 4 of my notes for Lecture 18) for the signature of an 8-dimensional closed manifold by testing it on \CP^4 and \CP^2 x \CP^2.

HW10, due Th, April 9: Read Lectures 38,39, bearing in mind that I will follow the material rather losely: the extraordinary part of Lecture 39 has been already discussed, the Atiyah-Hirzebruch spectral sequence has been presented in the lectures in a more abstract setting, and in the proof of the Bott periodicity theorem, I will follow Bott's original approach (following Milnor's "Morse theory").
1. Consider the following (coexact) Puppe sequence of a pair (X,A): A \subset X \subset (X\cup CA) \to (X\cup CA)/X=\Sigma A \subset \Sigma X ... Apply the functor \pi (-, W) of homotopy classes of maps to W, and show that the resulting sequence of maps and sets is "exact" in the following sense: (i) composing any two cosecutive arrows with a map to W yields a homotopy constant map; (ii) if composing an arrow with a map to W yields a homotopy constan map, then the map descends to a map of the next space in the sequence.
2. Prove that complex cobordism ring \Omega^*_U(\CP^n)=\Omega^*_U(pt)[x]/(x^{n+1}), where x is the Thom class of the (dual to the) tautological line bundle over \CP^{n-1}.
3. Intersection index of (say) complex bordism classes of a (stably almost) complex manifold X takes values in the bordism ring of the point, and is defined by "multiplying" (as in Problem 2 of HW9) the dual cobordism classes, and taking the absoite bordism class of the result (i.e. forgetting X). Use this structure to compute the class A+Bx+Cx^2 \in \Omega^*_U(\CP^2) of a degree-d non-singular complex curve C in \CP^2, where x is Poincare-dual to the line \CP^1 in \CP^2 (as in the previous exercise), and (A,B,C are polynimials in the generators \CP^1,\CP^2,... over the rationals in the bordism ring of the point).
4. In the group Diff_1(\Z_2) of diadic diffeomorphosms of the line, compose x=u+x_1u^2+x_2u^4+... with y=u+y_1x^2+y_2u^4+..., and then compute coefficients D_1,D_2,D_3 of f(x \circ y) = f+D_1f y_1+D_2f y_1^2+D_3f y_1^3+... (which are the left-invariant differential operators on the group, representing Sq^1, Sq^2, and Sq^3 respectively). Verify the relation Sq^1Sq^3=Sq_2^2 by checking that D_1 D_3=D_2^2 (and remembering that 1+1=0).

Some references:
W. Massey. Exact couples in algebraic topology, Parts I and II. Annals in Math., Second ser., vol. 56, No. 2 (1952), pp. 363-396. Available through JSTOR. Link to an excerpt
A. Dold. Relations between ordinary and extraordinary homology. Colloquium on Algebraic Topology, August, 1962, Aarhus Universitet. (7 pages)
D. Quillen. On the formal group law of unoriented and complex cobordism theory. Bull. AMS 75 (1969), 1293-1298. link