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A Brief Introduction to Morse Theory Gianmarco Molino A Brief Introduction to Morse Theory Definitions Motivating Example First Results Gianmarco Molino Morse Inequalities University of Connecticut Existence Results October 27, 2017 Applications and Further Reading

What is Morse Theory? A Brief Introduction to Morse Theory Gianmarco Molino In the following, let M be a closed, n -dimensional smooth Definitions manifold. Motivating Example Initiated by Marston Morse, 1920-1930. First Results Study of critical points of smooth functions f : M → R . Morse Inequalities Attempts to recover topological information about M . Existence Results Applications and Further Reading

A Brief Introduction to Morse Theory 1 Definitions Gianmarco Molino 2 Motivating Example Definitions Motivating Example 3 First Results First Results Morse 4 Morse Inequalities Inequalities Existence Results 5 Existence Results Applications and Further Reading 6 Applications and Further Reading

Definitions A Brief Introduction to Morse Theory Gianmarco Molino A smooth manifold M is a topological manifold with Definitions compatible smooth atlas. Motivating A critical point p ∈ M of a smooth function f : M → R is Example a zero of the differential df . First Results Morse The Hessian H p ( f ) of f at a critical point p ∈ M is the Inequalities matrix of second derivatives. (Independent of coordinate Existence Results system at critical points.) Applications and Further Reading

Morse Functions A Brief Introduction to Morse Theory Gianmarco Molino A smooth function f : M → R is called Morse if its critical Definitions points are isolated and nondegenerate (that is, the Hessian Motivating of f is nonsingular.) Example Remark: Nondegenerate critical points are necessarily First Results isolated. Morse Inequalities The index λ ( p ) of a critical point p is the dimension of the Existence negative eigenspace of H p ( f ). Results Applications and Further Reading

Torus with height function A Brief Introduction Consider the 2-dimensional torus T 2 embedded in R 3 and a to Morse Theory tangent plane: Gianmarco Molino Definitions Motivating Example First Results Morse Inequalities Existence Results Applications and Further Reading Define f : T 2 → R to be the height above the plane.

A Brief The function h has 4 critical points, a , b , c , d , with Introduction to Morse λ ( a ) = 0 , λ ( b ) = λ ( c ) = 1 , λ ( d ) = 2. Theory Gianmarco Molino Definitions Motivating Example First Results Morse Inequalities Existence Results Applications and Further Reading

Morse Lemma A Brief Introduction to Morse Theory Gianmarco Molino Definitions Nondegeneracy of critical points is a generalization of Motivating non-vanishing of the second derivative of functions Example f : R → R . First Results We thus expect to be able to describe M in relation to Morse Inequalities these points. Existence Results Applications and Further Reading

Morse Lemma A Brief Introduction to Morse Theorem (Lemma of Morse) Theory Gianmarco Let f ∈ C ∞ ( M , R ) , and let p ∈ M be a nondegenerate critical Molino point of f . Then there exists a neighborhood U ⊂ M of p and Definitions a coordinate system ( y 1 , . . . , y n ) on U such that y i ( p ) = 0 for Motivating all 1 ≤ i ≤ n , and moreover Example First Results f = f ( p ) − ( y 1 ) 2 − · · · − ( y λ ) 2 + ( y λ +1 ) 2 + · · · + ( y n ) 2 Morse Inequalities Existence where λ = λ ( p ) is the index of p. Results Applications and Further Corollary Reading If p ∈ M is a nondegenerate critical point of f , then it is isolated.

A Brief Introduction to Morse Theory Gianmarco Given f : M → R , define the ‘half-space’ Molino M a = f − 1 ( −∞ , a ] = { x ∈ M : f ( x ) ≤ a } . Definitions Motivating Example First Results Theorem (Milnor) Morse Inequalities Let f : M → R be C ∞ . If f − 1 ([ a , b ]) is compact and contains no critical points of f , then M a is diffeomorphic to M b and Existence Results furthermore M a is a deformation retract of M b . Applications and Further Reading

A Brief Introduction to Morse Theory Gianmarco Molino The gradient of f induces a local 1-parameter family of Definitions diffeomorphisms φ t : M → M away from critical points. This Motivating allowing the points of M a to flow along these gives the desired Example First Results deformation retract. Morse Remark: The condition that f − 1 ([ a , b ]) be compact cannot be Inequalities relaxed. Existence Results Applications and Further Reading

A Brief Introduction to Morse Theory Gianmarco Molino Theorem (Milnor) Definitions Let f : M → R be C ∞ and let p ∈ M be a (nondegenerate, Motivating isolated) critical point of f . Set c = f ( p ) and λ = λ ( p ) to be Example First Results the index of p. Suppose there exists ǫ > 0 such that Morse f − 1 ([ c − ǫ, c + ǫ ]) is compact and contains no critical points of Inequalities f other than p. Then for all sufficiently small ǫ , M c + ǫ has the Existence homotopy type of M c − ǫ with a λ -cell attached. Results Applications and Further Reading

A Brief Introduction to Morse Theory Gianmarco Molino The key observation is that when crossing a critical point, the Definitions Morse Lemma is applicable. It can be shown that attaching a Motivating Example λ -cell e λ to M c − ǫ along the ( y 1 , . . . , y λ ) axis, First Results M c − ǫ ∪ e λ ∼ Morse = M c + ǫ . Inequalities Existence Results Applications and Further Reading

A Brief Introduction to Morse Theory Gianmarco Molino Intuitively then, the manifold can be constructed from cells determined by the indices of the critical points. Definitions Motivating Example Theorem (Milnor) First Results If f : M → R is Morse and for all a ∈ R it holds that M a is Morse Inequalities compact, then M has the homotopy type of a CW complex Existence with one cell of dimension λ for each critical point with index λ . Results Applications and Further Reading

A Brief Introduction to Morse Theory Gianmarco Molino This is enough to get a few results. For example, Definitions Theorem (Reeb) Motivating Example Let M be a compact smooth manifold, and let f : M → R be First Results Morse Morse. If f has only two (nondegenerate) critical points, then Inequalities M is homeomorphic to a sphere. Existence Results Applications and Further Reading

Differential Forms A Brief Introduction to Morse Theory Gianmarco Molino Recall the space Ω k ( M ) of differential k -forms over M , and the Definitions exterior derivative d : Ω k → Ω k +1 , which gives rise to the Motivating Example deRham co-chain complex First Results Morse 0 → · · · d → Ω k ( M ) d → Ω k +1 ( M ) d Inequalities − − − → · · · → 0 Existence Results Applications and Further Reading

Betti Numbers A Brief Introduction to Morse Theory The associated cohomology group is the deRham cohomology Gianmarco group Molino dR ( M ) = ker d : Ω k → Ω k +1 Definitions H k Motivating im d : Ω k − 1 → Ω k Example First Results and further we define the k -th Betti number of M , Morse Inequalities β k = dim H k dR ( M ) . Existence Results This cohomology encodes topological information about the Applications and Further manifold algebraically, and is the starting point for fields such Reading as Hodge Theory and Index Theory.

The Betti numbers are topological invariants. They are related A Brief Introduction to the classical Euler characteristic χ ( M ) by to Morse Theory n Gianmarco � ( − 1) k β k . Molino χ ( M ) = k =0 Definitions Motivating Which is an explicit expression for the following lemma from Example Index Theory: First Results Morse Lemma Inequalities Let D = d + δ be the Dirac operator for the Hodge Laplacian Existence Results ∆ = D 2 = d δ + δ d. Then Applications and Further Reading χ ( M ) = index( D ) where index( D ) = dim ker( D ) − dim coker( D ) denotes the analytic index.

A Brief Introduction to Morse Theory Gianmarco Molino Definitions Unfortunately, the Betti numbers can be remarkably difficult to Motivating Example compute directly. This is where Morse Theory provides a First Results solution. Morse Inequalities Existence Results Applications and Further Reading

Weak Morse Inequalities A Brief Let f : M → R be Morse, and define the Morse numbers, M k , Introduction to Morse by Theory M k = # { p ∈ M , df ( p ) = 0 , λ ( p ) = k } Gianmarco Molino Theorem (Weak Morse Inequalities) Definitions Motivating Let M be compact, β i be the Betti numbers of M, f : M → R Example be Morse, and M k be the Morse numbers of f . Then First Results Morse Inequalities β k ≤ M k Existence Results and moreover Applications and Further Reading n n � ( − 1) k β k = � ( − 1) k M k . χ ( M ) = k =0 k =0