Morse Theory Problems
More to be added as the quarter progresses! Find a few that interest you and give them
a try.
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Flows, regular interval theorem
1. Suppose M is a smooth manifold without boundary, X a smooth vector field on M, and f
a smooth real-valued non-vanishing function on M. Let Y=fX. Show that X and Y have the
same trajectories (a trajectory is the image of an integral curve, as opposed to the integral
curve itself).
2. Recall that the regular interval theorem is false without the assumption f −1 [a, b]
compact. (There are trivial counterexamples; just remove a point from the interior of a
cylinder as in Milnor top of p. 14, or an even simpler analogue for 1-dimensional manifolds.)
Suppose, however, that f is a proper smooth function on M and [a, ∞) is a regular interval
with f −1 a nonempty. Then
a) f −1 [a, ∞) is diffeomorphic to f −1 a × [0, ∞), and f −1 (−∞, a] is a deformation retract
of M . (Compare problem 13.25 in Lee’s Smooth Manifolds.)
b) M is diffeomorphic to the interior of a compact manifold with boundary.
c) M is homotopy-equivalent to a compact manifold with boundary.
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Morse functions
1. Show that f : M −→R is a Morse function if and only if the differential df : M −→T ∗ M is
transverse to the zero section. (T ∗ M is the cotangent bundle.)
2. Let M ⊂ Rm be a closed smooth embedded submanifold. Then for almost all v ∈ S m−1 ,
the height function hv (x) = hv, xi is a Morse function on M .
Suggested outline: Define F : M × (Rm )∗ −→T ∗ M by restriction to tangent spaces, i.e.
F (x, λ)(v) = λ(v) for v ∈ Tx M . Show that F is smooth and a submersion. Then apply the
parametric transversality theorem ([Lee], Theorem 6.35) together with Exercise 2 above.
3. Sometimes it is convenient to have all the critical points at distinct levels, i.e. if p, q
are critical then f (p) 6= f (q). In this problem you’ll show this can always be arranged. Part
of the problem is to make the statements below more precise.
a) Level-adjusting lemma. Let f be a Morse function on M , p a critical point. Let ρ
be a suitable bump function supported on a small neighborhood U of p and for > 0 let
g = f + ρ. Then for sufficiently small , g is a Morse function with the same critical points
and indices as f (and in fact equal to f away from the support of ρ).
b) Now show that there is a Morse function g with the same critical points and indices
as f , but with distinct critical values.
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Remark: Even nicer would be to arrange that the critical points occur in order of increasing index, ι(p) < ι(q) implies f (p) < f (q). This is much harder and constitutes one key
step (of many) in the proof of the h-cobordism theorem. Note that the exercise doesn’t help
because of the indeterminately small in the level-adjusting lemma.
4. Give a simple pictorial argument (no rigor, please!) “showing” that S 2 admits Morse
functions with an arbitrarily large number of critical points. Unless you are prone to hallucination, your examples should always have an even number of critical points. This phenomenon will be explained later (the number of critical points, that is, not anyone’s tendency
to hallucinate). In the meantime, perform the following thought experiment: Try to construct, pictorially or otherwise, a Morse function on S 2 with exactly three critical points.
Why doesn’t it work?
Note: Proceed as you see fit, but what I have in mind by “pictorial argument” is to use
the fixed height function f (x, y, z) = z, but vary the way S 2 is embedded in R3 .
5.
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Cell structures
1. Let M be compact, f a Morse function on M. A vector field X on M is gradient-like if
(i) for all non-critical points p, Xp f > 0; and (ii) there are Morse chart coordinates near
each critical point p such that X(x1 , ..., xn ) = (−x1 , −x2 , ..., −xk , xk+1 , ..., xn ), where k is the
index of p.
a) Show that gradient-like vector fields exist (make use of a Riemannian metric induced
by Morse charts near critical points).
In the rest of the problem, assume a gradient-like X has been fixed.
b) Let θ denote the flow associated to X. Show that for any x ∈ M , the limit of θ(t, x)
as t−→∞ exists, and is a critical point of f .
We call this critical point L+ (x). Similarly, the limit as t−→ − ∞ exists and is a critical
point denoted L− (x).
Now fix a critical point p of index k. The inset Win p of p is the set of all points that
flow to p - that is, Win p = {x ∈ M : L+ (x) = p}. Similarly, the outset Wout p is the set of all
points that came from p at the beginning of time - that is, Wout p = {x ∈ M : L− (x) = p}.
c) Show that Win p is a smooth submanifold of dimension k, and in fact is diffeomorphic
to Rk . (At least show homeomorphic to Rk , which is easier.) Similarly for Wout p, with k
replaced by n − k.
d) Show that the intersection of Win p and Wout p is precisely {p}, and this intersection is
transverse.
Remark: Note that this gives a partition of M into cells, namely the insets of the critical
points, that corresponds exactly to the up-to-homotopy CW-decomposition given by the
Main Theorem. However, it is not at all apparent that this really is a CW-decomposition.
The problem is that we don’t have characteristic maps from the appropriate closed discs.
The outsets give a “Poincare dual” decomposition into cells. This problem will be important
in our sketch proof of Smale’s theorem.
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2. Morse theory tells us much more than that every smooth manifold has the homotopytype of a CW-complex, in that we get specific information relating the dimensions of the cells
to the indices of critical points. There is a simpler way, however, to just get the homotopyequivalence with a CW-complex (this assumes you know about tubular neighborhoods): By
considering suitable subdivisions into cubes, show that every open subset of Rn is a CWcomplex. Since every smooth M can be embedded as a closed submanifold of a Euclidean
space, and is a deformation retract of a tubular neighborhood, this does the job.
Remark: In fact every smooth manifold can be triangulated. This is much harder to
prove, and despite being a theoretically stronger statement than “M has the homotopy-type
of a CW-complex”, does not have anything like the power of Morse theory.
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Focal points
1. Here’s one that will give you a good feeling for focal points. Let M be a torus in R3 ,
given as a surface of revolution; explicitly, take a circle of radius r in the xz-plane centered
at (a, 0, 0) with a < r and rotate around the z-axis.
a) Show that the set of focal points of M is the union of the “core” circle of radius a in
the xy-plane and the z-axis, and that every focal point has multiplicity one. (Visually this
is very plausible; with a little patience and exploitation of symmetry you can compute the
relevant differential and make it rigorous.)
b) Let p = (e, 0, 0) with e > a + r, so in particular p is non-focal and Lp is a Morse
function. Find the critical points explicitly (this is trivial; there are four) and compute their
indices by counting focal points according to the recipe given in Milnor and in class.
Note: Since every focal point has multiplicity one, you literally just count them. Remember, however, that you don’t count all focal points on a given segment qp;
~ you only count
the q-focal points.
c) (A thought experiment). Any non-focal p yields a Morse function, so consider what
happens with other choices of p.
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