Math 265 - Differential Topology

Instructor: Ian Agol
Lectures: TuTh 2:00-3:30 pm, 205 Wheeler Hall (note: Th 1/20 will be in 891 Evans)
Course Control Number: 54547
Office: 921 Evans
Office Hours: TBA

Textbooks:
Morse Theory, John Milnor
Lectures on the h-cobordism theorem, John Milnor

Recommended:
Pits, Peaks, and Passes

An Invitation to Morse Theory, Nicolaescu

Riemannian Geometry, Petersen

Riemannian Geometry, Jost

Le theoreme du h-cobordisme, Cerf & Gremain

La stratification naturelle des espaces de fonctions differentiables ... , Cerf

Syllabus:
We'll discuss Morse theory, and its applications to
Bott periodicity and to the Poincare conjecture via the h-cobordism theorem.
We may discuss other related topics depending on time and interest, such as
exotic 7-spheres, Cerf theory, Morse theory for 3-manifolds, and Kirby
calculus.

Homework:
Homework 1, due Thursday 1/26:
1. take a torus of revolution in R^3, and tilt it until two saddle critical points have the same level.
Describe the cross-section.
2. which directions is the torus of revolution in Morse position with respect to height?
3. describe a function on the torus with only 3 critical points.

Homework 2, due Thursday 2/3:
Suppose M is a closed, orientable smooth 3-dimensional
manifold whose integral homology is isomorphic to the homology of S^3 and
f : M → R is a Morse function.
(a) Prove that f has an even number of critical points.
(b) Prove that a closed orientable 3-manifold with a Morse function with at most 4 critical points is
S^3, a lens space, or S^1 x S^2.

Homework 3, due Thursday 2/10:
Find the focal point of the hyperbola x^2-y^2=1 in R^2 at the point (1,0). Repeat
for the curve defined by the same equation in C^2. Use Morse
theory to determine the topological type of this curve.

Homework 4, due Thursday 2/17:
1. Let c : [0,∞) → S^n be a geodesic parametrized by arc length. For t > 0,
compute the dimension of the space of Jacobi fields X along c with X(0) =
0 = X(t). Use the Morse index theorem to compute the indices and nullities
of geodesics on S^n.

2. Prove that a geodesic is not length-minimizing past a conjugate point.

Homework 5, due Thursday 2/24:
1. Prove that the energy function is continuous on the space of loops (endowed with the metric
given at the beginning of chapter 16).

2. Prove that for a closed connected Riemannian manifold M, and points p, q in M which are not
conjugate, there are infinitely many geodesics connecting p and q.

Homework 6, due Thursday 3/3:
1. Let M be a symmetric space. Prove that the group <I_p : p ∈ M > generated by symmetries
through points of M is a closed (Lie) subgroup of Isom(M).

2. The rank of a geodesic γ is the dimension of the parallel vector fields E along γ such that R(E(t), γ'(t))γ'(t)=0 for all t. Prove that this is the nullity of the operator K_γ'(0)  when M is a symmetric space.

Homework 7, due Tuesday 3/29:

1. Compute Ω₁(4).
2. Ω SO(4) is homotopy equivalent to a cell complex K. Compute the number of cells of each dimension in the 4-skeleton K⁽⁴⁾( = K⁽⁵⁾ ).  Consider SU(2) with binvariant metric. Prove that the identity component of the isometry group of this metric is generated by the actions of SU(2) on itself by left- and right-multiplication. Use this to compute the double cover of SO(4) and its loop space.

Homework 8, due Tuesday 4/5:
Let (W; V, V') be a manifold triad.
1. Prove that W is an h-cobordism iff  V-> W and V' -> W are homotopy equivalences.
2. Prove that W is an h-cobordism if W, V, V' are simply-connected and H_*(W,V)=0.
3. Prove that a closed n-manifold M is homotopy equivalent to Sⁿ iff M is simply-connected and H_*(M)=H_*(Sⁿ).

Homework 9, due Tuesday 4/12:
Prove that  smooth homotopy n-spheres (n>5) modulo h-cobordism form a group under the operation of connect sum, with
identity the h-cobordism class of S^n.

Homework 10:

Prove that for a 5-dimensional simply-connected cobordism (W; V, V') with a Morse function with a single critical point of index 2, V'=V#(S^2 x S^2) or V'=V#(S^2 \tilde{x} S^2) (the twisted bundle).