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 Ω

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

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

3. Prove that a closed n-manifold M is homotopy equivalent to S

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).