Morse Theory A Rather Brief Introduction Johar M. Ashfaque Abstract What follows is a very basic introduction to Morse theory highlighting the key ideas.
Keywords: Morse Theory, Morse Polynomial, Poincaré Polynomial, Lacunary Principle.
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Key Definitions
A point α ∈ M is a critical point of f if
∂f
= 0, ∂xi x=α
∀i.
A critical point is non-degenerate if and only if the Hessian 2
∂ f
det 6= 0. ∂xi ∂xj x=α n×n The index k of a non-degenerate critical point α is the number of negative eigenvalues of the Hessian. Non-degenerate critical points are isolated. This is to say that there exists a neighbourhood of the critical point in which no other critical points of f are present. A smooth real-valued function on a manifold M is a Morse function if it has non-degenerate critical points.
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The Weak Morse Inequalities: The Idea
Suppose M is a compact differentiable manifold of dimension n. Further suppose f represents a smooth real-valued function on M that is f : M → R. Then the Morse inequalities constrains the number of critical points that the function f can have due to the topology of M. Let bk denote the k-th Betti number and mk denote the number of critical points of index k on the compact differentiable manifold M then bk ≤ mk . Note: The k-th Betti number of the manifold M sets a lower bound on the number of critical points of index k that the function f must have.
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The Weak Morse Inequalities
Define the Morse polynomial as Mt (M, f ) =
X
m k tk
and the Poincaré polynomial as Pt (M) =
X
dim(Hk (M))tk = 1
X
bk tk .
The Morse polynomial will always converge since it contains only a finite number of terms because the non-degeneracy makes the critical points discrete and the compactness of M allows only for a finite number of such points. We then have Mt (M, f ) ≥ Pt (M) X X ⇒ mk tk ≥ bk t k ⇒ m k ≥ bk ,
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k = 0, ..., n.
The Weak Morse Inequalities: The Strong Result
Setting ∆(t) = Mt (M, f ) − Pt (M) Morse found that for every non-degenerate function f there exists a polynomial Qt (f ) = q0 + q1 t + ... with non-negative coefficients such that Mt (M, f ) ≥ Pt (M) can be expressed in the form Mt (M, f ) − Pt (M) = (1 + t)Qt (f ).
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The Morse Lacunary Principle: Inequalities Become Equalities
Suppose that the Morse polynomial contains only even powers of t. Then Qt (f ) is the zero polynomial so that Mt (M, f ) = Pt (M). Proof: Let tk be first non-zero power f ). Then tk is also the first non-zero power in the Pin Mkt (M,k+1 difference ∆(t). But (1 + t)Qt (f ) = qk (t + t ) thus if Qt (f ) 6= 0 then tk+1 also occurs in the product and so too in Mt (M, f ) − Pt (M). This is not possible as tk+1 does not occur in Mt (M, f ) by our assumption and can not be in Pt (M) as it would violate the inequalities mk+1 ≥ bk+1 . Hence, Qt (f ) must vanish.
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An Example: The Height Function Of A Torus
Consider the function f : [0, 1] → R:
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It is worth noting that the height function of the torus is an example of a perfect Morse function. There are clearly 4 non-degenerate critical points for the height function of the torus. We only need to work out the index, k(α), for the 4 stationary points. It can be seen immediately that k(α) = 0 for the stationary point that appears at the minimum. k(α) = 1 for the two saddle points and k(α) = 2 for the stationary point at the maximum. The Morse polynomial can be evaluated simply for the torus to be Mt (M, f ) = 1 + 2t + t2 = (1 + t)2 . The first few Betti numbers are • b0 which denotes the number of connected components • b1 which denotes the number of holes • b2 which denotes the number of voids. Then Poincaré polynomial can be evaluated simply for the torus to be Pt (M) = 1 + 2t + t2 = (1 + t)2 .
The Morse polynomial and the Poincaré polynomial coincide as was expected.
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Kirwan’s Paper: Key Theorem [1]
Theorem: If the stationary phase approximation for f is exact then the index of every critical point in even. Hence f is a perfect Morse function (i.e. its Morse inequalities are in fact equalities). As an immediate consequence, we note that the dimension of M is even. Note: The statement of the theorem is equivalent to the lacunary principle.
A
Poincaré Polynomial And The Tori Betti numbers
b0
b1
Circle
1
1
Torus
1
2
1
3-Torus
1
3
3
1
4-Torus
1
4
6
4
Tori
b2
b3
b4
Poincaré Polynomial 1+t 1 + 2t + t2 = (1 + t)2 1 + 3t + 3t2 + t3 = (1 + t)3
1
1 + 4t + 6t2 + 4t3 + t4 = (1 + t)4
Table 1: This table gives the Betti numbers for the circle, torus, 3-torus and 4-torus with the corresponding Poincaré polynomials. From Table (1), we deduce that the Poincaré polynomial for the n-torus is given by (1 + t)n , and that the Betti numbers of the Poincaré polynomial are the binomial coefficients.
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References [1] F. Kirwan, Morse functions for which the stationary phase approximation is exact, Topology 26, 37 (1987).
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