## Fall-19. Math 215A (cnn 25351): Algebraic Topology

Instructor: Alexander Givental
Lectures: TuTh 3:30-5 in 70 Evans
Unofficial discussion section: Fr 3-5, 740 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
Grading policy: I promised to explain my 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 lower (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 everyone demonstrate good knowledge of the subject.

HW1, due Th, Sep. 5: Read Introduction (Lectures 1 and 2). Solve:
1. Show that a subset in \R^{\infty}:=\lim \R^n (the union of a nested sequence of euclidean spaces) is open if and only if the intersection of the subset with each of these euclidean subspaces is open.
2. Show that a compact subset in \R^{\infty} must lie in a finite-dimensional subspace \R^n.
3. Solve Exercise 1 from Lecture 2.
4. Solve Exercise 13 from Lecture 2 (and point out where you use the assumption that spaces X, Y are locally compact).

HW2, due Th, Sep. 12: Read Lecture 3 (perhaps less 3.5) and Lecture 6 about fundamental groups and coverings. Solve:
1. Exercise 1 in Section 6.2. (The catch is that the homotopy equivalence is not assumed to be in the category of base point spaces.)
2. Prove that the fundamental group of a loop space \Omega X of any base point space (X,x_0) is abelian. The same for any topological group. ( Hint: Start with the group T^2.)
3. Exercises 2 and 3 in Section 6.4.
4. Exercise 7 in Section 6.9. ( Hint: 185.)

HW3, due Th, Sep. 19: Read 7.1-7.2, Lecture 8 (perhaps plus sections 9.4, 9.7-9.9). Solve:
1. Exhibit a space with the fundamental group isomorphic to \Z_n (:=\Z/n\ZZ, the cyclic group of order n).
2. Exercise 8 from section 7.2.
3. Exercise 2 from section 8.2.
4. Exercise 16 from section 8.8.

HW4, due Th., Sep. 26: Read: Section 5.8, and Lecture 10 (perhaps without 10.5). Solve:
1. Prove that all fibers of a Hurewicz bundle over a path-connected base are homotopy equivalent to each other.
2. Compute all homotopy groups of \C P^{\infty} (infinite dimensional complex projective space).
3. Ecercise 15 of section 9.9.
4. Accepting the Inverse Function Theorem, derive that a smooth map \R^{m+n},0 --> \R^n,0 (defined in a nbhd of the origin), whose linear approximation (given by the Jacobi matrix) at the origin has rank n, is locally equivalent toits linear approximation (i.e. in suitable nonlinear local coordinate systems has the form (x_1,...,x_m, y_1,...,y_n) |--> (y_1,...,y_n)).

HW5, due Th, Oct. 3: Read Lectures 5 and 11. Solve:
1. Explain why Exercises 8 and 14 from Lecture 5 contradict each other, and give a counter-example to one of them.
2. This and the next question are about cell partitions of the grassmannian G(n,k). Show that a Schubert cell A is contained in the closure of Schubert cell B if and only if the Young diagram of A is contained in the Young diagram of B.
3. Suppose the Young diagram of B is obtained by adding one square to the Yound diagram of A (i.e. that dim B = dim A+1). Show that the (attaching) map from the boundary sphere of the cell B to the cell A modulo its boundary has degree (as a map between spheres of the same dimensions) equal 0 or 2 (+2 or -2 depending on the choice of the cells' orientations), and find out how the result depends on the position of the added square in the k x (n-k) rectangle.
4. Exercise 22 from Lecture 5.

HW6, due Th, Oct. 10: As I've mentioned in class, the topic for this week is "classification of principal and vector bundles". It is not adequately represented in the book (although it was fully represented in Fuchs' lectures on which the book is based). Some of the material can be found in sections 19.1, 19.2, 19.3 B, and 19.4 ABC.
For hw exercises of this week, solve:
1. Compute \pi_2 (G(n,k)) (the 2nd homotopy groups of grassmannians).
2. Exercises 5 and 6 from Section 11.5 (S^2 and S^3 x CP^{\infty} have the same homotopy groups but are not WHE, and the same for S^m x RP^n and S^n x RP^m unless m=n).
3. Exercise 9 from section 11.7.
4. Exercise 15 from section 11.9

HW7, due Tuesday (!) Oct. 22: Read Lecture 12 and probably 13.1-13.5. Solve:
1. Show that G_{+}(\infty, 2) is homotopy equaivalent to CP^{\infty}.
2. Prove that a group homomorphism G --> G' which is a weak homotopy equivalence induces a weak homotopy equivalence BG --> BG'.
3. Exercise 11 in 19.2 (the cotangent vector bundle of a projective n-space is stably equivalent to the sum of n+1 copies of the tautological line bundle).
4. Classify principal SL_2(\C)-bundles over \C P^2.

HW8, due Tue. Oct. 29: Read Lectures 13 and 14. Solve:
1. Exercise 2a from 12.2.
2. Exercise 1 from 13.1.
2. Exercise 11 from 12.5.
4. Compute (cellular) homology groups of the grassmannian G(4,2).

No seminar this Friday, Nov. 1 - sorry! - I will be away, in Philadelphia

HW9, due Tue. Nov. 5: Read Lectures 14,15. Solve:
1. Give an example of a non-contractible cell space with homology groups of a point.
2. Examine the proof of the Relative Hurevicz Theorem [resp. the Homological Whitehead Theorem] and argue that instead of simply-connectedness of the spaces X,A it suffices to assume that \pi_1(A) acts trivially in \pi_q(X,A) [resp. that X and Y are homotopy simple].
3. For n>1, obtain X_n from cylinder S^n x [0,1] by identifying S^n x 0 and S^n x 1 by a degree-2 map. Compute \pi_n(X_n). Show that embedding S^1 into X_n (as base point x [0,1]/0=1) induces isomorphism in homology (but not in homotopy).
4. Let X be a finite cell space. Prove that the alternate sum #_0-#_1+#_2-... of the numbers #_d of d-dimensional cells of X is a homotopy invariant of X (called the Euler charactristic . Namely, prove that it coincides with the alternated sum h_0(X;F)-h_1(X;F)+h_2(X;F)-... where h_q(X;F)=\dim H_q(X;F), the dimension of the homologogy group with coefficients in a field F.

HW10, due Tue, Nov. 12: Read Lectures 15,16. Solve:
1. Let F_1 -> F_0 be a 2-term complex of free abelian groups with 0 kernel and cokernel A. Let 0 -> G_1 -> G_0 -> B -> 0 be a short exact sequence of abelian groups. Tensor this sequence with the above 2-term complex to obtain a short exact sequence of 2-term complexes. Write out the (not-too-)long exact sequence of homology groups corresponding to this exact sequence of complexes. Derive that Tor(A,B)=Tor(B,A).
2. Let X be obtained from a sequence of cylinders C_n=S^1 x I by attaching the left end of each C_n to the right end of C_{n-1} by a degree-2 map S^1->S^1. Compute homology and cohomology of X with integer coefficients, and compare the result with the corresponding universal coefficient formula.
3. Check that in the cellular chain complex of D^m x D^n, the boundary operator satisfies d (D^m x D^n) = (d D^m) x D^n + (-1)^m D^m x (d D^n), and conclude from this that the cellular chain complex of the product of two cell spaces in the tensor product of their cellular chain complexes.
4. Exercise 20 of Lecture 15: Compute the integer coefficient homology of RP^2 x RP^2 using Kunneth's formula.

HW11, due Tue, Nov. 19:We'll begin the homology of manifolds, so start reading Lecture 17, e.g. 17.1 and 17.4, but keep in mind that I will follow it rather loosely. Solve:
1. Exercise 3 from 16.2.
2. Find out for which k and l there is non-zero degree map from S^k x S^{n-k} to S^l x S^{n-l}.
3. Use my conventions in the definition of cap-product and figure out correct signs in the Leibnitz rule.
4. Use Morse theory to show that a Morse function on RP^n cannot have fewer than n+1 critical points, and exhibit a Morse function with exactly n+1 critical points.

HW12, due Tue, Nov. 26: Our subject will be intersection theory and its applications, so read the material of Lecture 17 accordingly. Solve:
1. Let E be the total space of a real oriented n-dimensional vector bundle over a compact connected oriented m-dimensional manifold M, and E'=E\cup \infty denote the one-point compactification of E. Show that H_m(E,dE)=\Z (where dE is non-empty whenever dM is nonempty), H^n(E',\infty)=\Z, that the generators are represented respectively by the fundamental cycle of M and by the cocycle taking value 1 on the fundamental class of the (compactified) fiber, and that these two generators are related by the Poincare isomorphism. Don't use any conclusions of intersection theory, but rather rely on the definitions of fundamental classes, cap-products, etc.
2. Compute the self-intersection index of the real unit sphere inside the hypersurface z_1^2+...+z_n^2=1 in \C^n. Use the complex orientation of \C^n, i.e. the right-oriented basis (e_1, ie_1,...,e_n, ie_n), where (e_1,...,e_n) is a complex basis of \C^n.
3. Compute the cohomology ring of the complex manifold of complete flags V^1\subset V^2 in \C^3. ( Suggestion: Describe the ring by relations in the generators which come from the natural embedding of the flag manifold into \CP^2 x \CP^2*.)
4. Use intersection theory to prove the classical Borsuk-Ulam theorem: n odd continuous functions on S^n have a common zero. (A function f: S^n -> \R is odd , if f(-x)=-f(x) for all x in S^n.)

HW13, due Tue, Dec. 3: Read section 17.2 about (co)comology of a non-orientable manifold with orienting coefficients, and
1. Using the induction on attaching handlebodies, construct the fundamental class of the n-fold X lying in H_n(X, \Z_T).
2. Using the same induction, establish Poincare isomorphisms between integer homology and cohomology with coefficients in \Z_T.
3. Using these, extend Lefschetz' fixed point formula to the cace of possibly non-orientable manifold.
4. Suppose a smooth map f: \R^n -> \R^n has an isolated, but possibly degenerate fixed point at the origin. Show (e.g. using Sard's lemma) that f can be perturbed in an \epsilon-neighborhood of the origin to have only non-degenerate fixed points inside the neighborhood, and remain unchanged outside. Prove that the index of the fixed point (defined as the degree of the map (f(x)-x)/|f(x)-x| between n-1-dimensional spheres) coincides with the sum of the indices +1/-1 of the newly born non-degenerate fixed point.

HW14, due Tue, Dec. 10: The material to consult in the book is probably 19.4 D,E and 19.5, but you should keep in mind that the approach there is based on obstruction theory, which we have skept for the time being. Solve:
1. Prove that a real vector bundle is orientable if and only if its 1st Stiefel-Whitney class w_1=0.
2. Prove that cohomology ring H^*(Fl_n) of the manifold of complete flags 0=V^0\subset V^1\subset ... \subset V^n=\C^n is \Z[x_1,\dots x_n]/(elem. symmetric functions of x_1,... x_n), where x_i are the 1st Chern classes of the line bundles V^i/V^{i-1}.
3. Compute Chern classes of the tangent bundle of \C P^n and Stiefel-Whitney classes of the tangent bundle of \R P^n.
4. Show that if a closed 2-dimensional manifold M is the boundary of a compact 3-dimensional manifold, then < w_2(T_M), [M]>=0, and derive from this that M=\R P^2 is not the boundary of any 3-dimensional manifold.

Final Exam (please submit your solutions via email by Tue, Dec. 17) and its solutions!