Spring-2024. Math 215B (cnn 26023): Algebraic Topology

Instructor: Alexander Givental
Lectures: TuTh 9:30-11 in 740 Evans
Plan: We will loosely follow Homotopical Topology by A. Fomenko - D. Fuchs (Graduate Texts in Mathematics), 2nd ed., 2016, available to our library patrons in electronic format
In contrast with Hatcher's "Algebraic Topology" (used in the Fall semester and covering fundamental groups and coverings - sorry about the tautology - and cohomology theory first and only then digging into homotopy groups), this book treats homotopy groups first. So, we'll start with a brief overview of the material (mostly covered in 215A) of Lectures 1-7 of Fomenko-Fuchs text (classical spaces and constructions, homotopy equivalence, CW-complexes, fundamental groups, coverings and the van Kampen's theorem), and then study the material of Lectures 8-11 followed by applications to the classification of principal and vector bundles. Then an overview of (co)homology theory from 215A (corresponding to Lectures 12,13,15,16 of Fomenko-Fuchs) will be given followed by Morse theory, intersection theory on manifolds, and the rest of Chapter 2 (obstructions, charactersitic classes, etc.) Then the plan is to continue with the material of Chapter 3: spectral sequences and their applications, and then ... intro to equivariant cohomology? to K-theory? to cobordisms? we'll see.
Homework: due weekly on Gradescope by Tue 1 a.m.
Office hours: Th 12:30-3 pm in 701 Evans
Grading policy: Here is my proprietary one which I tried successfully in some courses and (as some requested) should use this time too. The starting point is: 60% homework + 40% take-home final (most likely during the RRR week). Next: each individual weekly hw score which is lower (percentage-wise) than your score on the final will be dropped - together with its weight. E.g.: if 1/3 of your hws is below your final score and 2/3 are above, then your total score becomes composed 50% of your final score and 50% 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.

HW1, due Tue, Jan. 23: Read Lectures 1,2,3,5. Solve:
1. Which of the following spaces are pairwise homeomorphic and which are not:
(a) the orthogonal group SO_3, (b) the space T_1S^2 of unit tangent vectors to S^2, (c) the Stiefel manifold V(3,2), (d) in the complex 3-space \C^3 with coordinates z_1,z_2,z_3, the set of unit vectors (i.e. |z_1|^2+|z_2|^2+|z_3|^2=1) satisfying z_1^2+z_2^2+z_3^2=0, (e) the real projective space \RP^3, (f) S^2 x S^1 ?
2. In the subsection 5.4D (pages 48-49) the partition of flag manifolds into cells (known as Bruhat cells ) is defined.
(a) For the manifold of complete flags V^1\subset V^2\subset \C^3 in (say, complex) 3-space (actually the field doesn't matter), describe explicitly which flags belong to which cell (the total number should be 3!)
(b) Let c_k be the number of k-dimensional (in complex units) Bruhat cells in the manifold of complete flags in an n-dimensional complex space. Show that the generating function c_0+c_1q+c_2q^2+... of this sequence is equal to the "q-factorial": product of (1-q^k)/(1-q) over k=1,...,n, and check this for your answer in (a).
Hint: Consider the Bruhat cell partition of the manifold of complete flags in the space F^n, where F is a finite field with q elements.

HW2, due Tue, Jan. 30: Read Lecture 8 (and review Lectures 6,7, the material of which should be familiar from 215A). Solve:
3. Prove that a CW-complex X is contractible if it is the union of nested subcomplexes X_1\susbet X_2\subset ... such that each X_n is contractible inside X_{n+1} (and conclude that the infinite dimensional sphere S^{\infty} is contractible ).
4. Exercise 8 from Lecture 8: Prove that the image of the homomorphism from \pi_2(X,x_0) in \pi_2(X,A,x_0) lies in the center of the latter group.

HW3, due Tue, Feb. 6: Read Lecture 9. Solve:
5. Let X be the space obtained by attaching one end of a 2-dimensional cylinder to the other end by the double cover map of the circle. (This space is sometimes called the "mapping torus" of the doble-cover map of S^1 to S^1.) Compute all homotopy groups of X.
6. Prove that fibers of a Hurewicz fibration over a path-connected base are homotopy equivalent to each other.

HW4, due Wed, Feb. 14: Read Lectures 10, 11. Solve:
7. Compute \pi_2(S^1 V S^2) (the 2nd homotopy group of the bouquet of S^1 and S^2) and the action of the fundamental group on it.
8. Compute the 3-rd homotopy group of unitary groups U_n for all n. Hint: Use the fibration of U_2 over U_1 defined by the determinant function, and the fibrations of U_n over S^{2n-1} defined by projection of a unitary matrix to its 1st column.

HW5, due Tue, Feb. 20: Read Lecture 11. Next week we will study an application of homotopy theory to the classification of principal (not in Hatcher's sense) and vector bundles. We won't exactly follow any text, but the material of 19.1,19.2, 19.4AB can serve as a weak approximation. Solve:
9. Prove that S^2 and S^3 x \CP^{\infty} have isomorphic homotopy groups but are not weakly homotopy equivalent.
10. (Exercise 14 on page 140.) Let X be a K(G, n), and Y be a cellular K(H,n). Show that a map from Y to X inducing a given group homomorphism \phi from H to G exists, and is unique up to homotopy.

Some asked for a written source on the classification theory of prinicpal and vector bundles. If you really need one, I can suggest "Beginner's course in topology: Geometruic chapters" by Fuchs and Rokhlin. Beware though that this is a different style text, more a reference than a textbook.

HW6, due Tue, Feb. 27: The plan is to begin a review of (co)homology theory and to cover during this week the material of Lectures 12,13 together with that of Lecture 14, which provides a relation between homology and homotopy groups. (Beware: formulations of Relative Hurewicz's and Whitehead's theorems in Fuch's book are utterly messed up!) Solve:
11. Show that the tautological embedding of the infinite dimensional projective space \C P^{\infty} into the grassmannian G_{+}(\infty,2) of oriented 2-dimensional subspaces in \R^{\infty} is a homotopy equivalence. (If it is not obvious to you what embedding I am talking about, it is also your job to figure this out.)
12. Given a continuous group homomorphism f from G to G' construct a map from BG to BG', which is a weak homotopy equivalence whenever f is. Hint: To be more accurate, the existence of the map is not guaranteed even if G=G' unless BG is cellular, but under this (or some other suitable) assumption construct a map between the entire fibrations which is fiberwise identical to f.

HW7, due Tue, Mar. 5: Read (review?) the material of Lecture 15 (cohomology, universal coefficients, Kunneth formula) and Lectures 20,21 on spectral sequences. Solve:
13. Classify principle SL_2(\C)-bundles over \C P^2.
14. Let X be the mapping torus of a degree-2 map of (S^2,pt) to itself (i.e. obtained from the cylinder S^2 x [0,1] by gluing its boundaries by this degree-2 map). Show that the embedding of S^1 into X as the interval pt x [0,1] glued at the endpoints induces isomorphism in homology but is not a homotopy equivalence.

HW8, due Tue, Mar. 12: Read Lectures 22 and 23.1-23.3 (about spectral sequences of fibrations) and review the material of 16.1-16.4 (about multiplications). Solve:
15. Prove that each skeleton sk_n X of a contractible CW-complex X is homotopy equivalent to a bouquet of n-spheres. (Warning: This is not necessarily sk_n X/sk_{n-1} X.)
16. Compute the cohomology of the universal cover of the mapping torus space from problem 14.

HW9, due Tue, Mar 19: The plan for the material before the break is to (a) discuss Thom isomorphisms and intersection theory on manifolds (but I won't really follow the material of Lecture 17),(b) multiplication in cohomological spectral sequences of fibrations (Lecture 24), (c) applications to cohomology of classifying spaces BU_n, BSp_n, BSO_n, BO_n, and (d) characteristic classes of vector bundles and their properties. Or, maybe, not that fast - we'll see. Note the change in the hw assignment! Solve:
17. A fibetwise map f: E\to E' between (homologically simple) fibrations \pi: E\to B and \pi': E'\to B' inducing a celluar map g: B\to B' between CW-bases respects the filtrations of the total spaces by E_p and E'_p, and hence defines a morphism betwen the spectral sequences of the fibrations, with the homomorphisms (f_*)^2_{pq}: H_p(B;H_q(F))\to H_p(B';H_q(F') induced by g and the restriction F\to F' of f to the fibers. Prove that even when the map g: B\to B' is not cellualar, the morphism between the spectral sequences is well-defined starting the term E^2.
Suggestion: Consider the fibration \E \to B x [0,1] (with the fiber F') induced by a homotopy B x [0,1]\to B' between g_0=g and its cellular approximation g_1, and examine the morphisms between spectral sequences induced by fibered maps: E\to \E_0 (over B x 0), \E_1\to \E, and \E_1 \to E' (all inducing cellular maps id_B, B x 1\subset B x [0,1] and g_1: B x 1 \to B' between the respective bases).
Remark. For Hurewicz fibrations, the construction could be simpler due to the Covering Homotopy Property valid for maps of arbitrary (rather than cellular) spaces.
18. Let M be a closed oriented n-dimensional manifold embedded smoothly into an (oriented) sphere S^N together with its tubular neighborhood U. Let \t \in H^{N-n}(U,dU) be the Thom class of the normal bundle of M in S^N considered as an element of H^{N-n}(S^N,S^N-U) via excision. Prove that the cap-product of the generator [S^N] in H_N(S^N) with this Thom class is a fundamental class of M, i.e. that for every point x in M it reduces to the generator in H_n(M,M-x).

HW10, due Tue, Apr. 2: Read Lecture 18: Obstruction theory. Solve:
19. Let (x_1,x_2,x_3) be coordinates on \C^3 and (y_1,y_2,y_3) be coordinates on the dual \C^3. Identify the hypersurface F in \C P^2 x \C P^2 given by the equation x_1y_1+x_2y_2+x_3y_3=0 with the manifold of complete flags in \C^3. Show that the homomorphism from H^*(\C P^2 x \C P^2)=\Z[u,v]/(u^3,v^3) to H^*(F) induced by the inclusion is surjective and find its kernel. Hint: A possible approach is to use intersections of submanifolds in lieu of cohomological cup-product.
20. Show that the tangent bundle T to \R P^n can be described as Hom(L, \R^{n+1}/L) where L is the tautological (M\"obius) line bundle, use this to compute Stiefel-Whitney classes of T, and conclude that they all are zeroes when n+1 is a power of 2.

HW11, due Tue, Apr. 9: Read 19.3-19.5, and Lecture 24 (multiplication in spectral sequences). Solve:
21. Show that composition of loops \Omega K(G,n+1) x \Omega K(G,n+1) \to \Omega K(G, n+1) induces the usual addition operation in cohomology H^n(X;G)=\pi(X, K(G,n))=\pi(X; \Omega K(G,n+1)) of a CW-complex X.
22. Show that the 1st Chern class of a complex n-dimensional vector bundle V over a CW-complex B (defined in terms of the obstruction theory) coincides with the 1st Chern class of its ``determinant'' line bundle \wedge^n V (the top exterior power of V).

HW12, due Tue, Apr. 16: The plan for the 3rd week of April is to discuss equivariant cohomology and fixed point localization, which is a very efficient computational tool in many applications. This subject is not represented in our textbooks, but a good (and concise!) source to which I will more-or-less follow is Chapter 3 in the book "Cohomology theory of topological transformation groups" by Wu-Yi Hsiang. Solve:
23. Let t\in H^k(E, E-B) be the Thom class of an oriented k-dimensional vector bundle E over a CW-complex B. Prove that the restriction of t to the zero section is the Euler class of the bundle. Derive from this that if E is a smooth fibration over a closed (oriented) manifold, then the fundamental class of zero locus s^{-1}(0) of a smooth section s transverse to the zero section is Poincare-dual in B to the Euler class of the bundle.
24. Use multiplication in the spectral sequence of the fibration over K(\Z,3) of its (contractible) path space (with the fiber K(\Z,2)) to compute the group H^6(K(\Z,3)) and identify in it the square e^2 of the fundamental class e\in H^3(K(\Z,3).

HW13, due Tue, Apr. 23: Here's my "lecture notes" for this week's material. Solve:
25. Show that H^*(\C G(n,k))=\Z[c'_1,...,c'_k, c'_1,...,c'_{n-k}]/I, where c'_i and c''_j are the Chern classes of the tautological \C^k- and \C^{n-k}=\C^n/\C^k-bundles, and the ideal I is generated by the homogeneous components of the relation (1+c'_1+...+c'_k)(1+c''_1+...+c''_{n-k})=1. Suggestion: First compute the U_n-equivariant cohomology as an H^*(BU_n)-algebra.
26. Let x_1,x_2,x_3 be the T^3-equivariant 1st Chern classes of line bundles V^1, V^2/V^1, \C^3/V^2 over the manifold F_3 of complete flags in \C^3. Write out the fixed point localization formula for the "integral" of a polynomial P(x_1,x_2,x_3) over F_3, and apply it in order to compute Chern numbers of F_3, i.e. the values on the fundamental class [F_3] of the Chern classes c_1^3, c_1c_2, c_3 of the tangent bundle. Suggestion: To find the T-equivariant Euler classes of the tangent spaces at each of the 6 fixed points, think of F_3 as the homogeneous space of GL_3(\C)/(upper-triangular matrices).

HW ANSWERS

HW14: The reading material for this week is 38.5 from Fomenko-Fuchs and/or 4.3 from Hatcher (since we are going to informally discuss extraordinary cohomology theories and spectra) and as much as you care from Lecture 44 of Fomenko-Fuchs on the cobordism theory. There will be no problem solving due this week, but over the weekend I will post/send the final exam text (which may refer to some of this week's material), and then you will have another week to submit your solutions. Respectively we won't have classes (nor office hours) during the RRR week.