An Introduction to Morse Theory Gianmarco Molino UConn Sigma - PowerPoint PPT Presentation

An Introduction to Morse Theory Gianmarco Molino UConn Sigma Seminar 27 July, 2017 Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 1 / 1024

A quick introduction to Differential Geometry Geometry is the study of shape, size, relative position of figures, and the properties of space, and has been historically one of the major motivating reasons for the field of mathematics. In applying the methods of calculus to this, we arrive at the modern field of Differential Geometry. Our primary objects of interest are manifolds. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 2 / 1024

A quick introduction to Differential Geometry A manifold is a generalization of Euclidean geometry, defined as objects that “locally look like” Euclidean R n . The first examples of a nontrivial manifold are surfaces in R 3 , such as spheres, tori, and surfaces of revolution. Some 2-dimensional manifolds can’t be embedded in R 3 , like the Klein bottle. We can also consider higher-dimensional manifolds, but it can be very hard to visualize these. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 3 / 1024

A quick introduction to Differential Geometry Moreover, since we are considering differential geometry, we want to consider smooth manifolds. Simply put, smooth manifolds are manifolds on which calculus can be done; there can be no sharp corners. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 4 / 1024

A quick introduction to Differential Geometry One idea to keep in mind: when working with manifolds, you have to jump back and forth between local and global properties. Locally (that is, in a small area around any point,) manifolds look just like R n . We have coordinate systems, and most of the ideas from multivariable calculus can be carried forward in the way you expect. Globally (that is, on the whole manifold,) most of these ideas don’t usually work. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 5 / 1024

What is Morse Theory? In the following, let M be a closed smooth manifold of dimension n . Initiated by Marston Morse, 1920-1930. Study of critical points of smooth functions f : M → R . Attempts to recover topological (not dependent on calculus) information about M . Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 6 / 1024

Definitions 1 Motivating Example 2 First Results 3 Morse Inequalities 4 Existence Results 5 Applications and Further Reading 6 Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 7 / 1024

Definitions A smooth manifold M is a topological manifold with compatible smooth atlas. A critical point p ∈ M of a smooth function f : M → R is a zero of the differential df . The Hessian H p ( f ) of f at a critical point p ∈ M is the matrix of second derivatives. (Independent of coordinate system at critical points.) Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 8 / 1024

Morse Functions A smooth function f : M → R is called Morse if its critical points are nondegenerate (that is, the Hessian of f is nonsingular.) ◮ Remark: Nondegenerate critical points are necessarily isolated. The index λ ( p ) of a critical point p is the dimension of the negative eigenspace of H p ( f ). Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 9 / 1024

Torus with height function Consider the 2-dimensional torus T 2 embedded in R 3 and a tangent plane: Define f : T 2 → R to be the height above the plane. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 10 / 1024

The function h has 4 critical points, a , b , c , d , with λ ( a ) = 0 , λ ( b ) = λ ( c ) = 1 , λ ( d ) = 2. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 11 / 1024

Morse Lemma Nondegeneracy of critical points is a generalization of non-vanishing of the second derivative of functions f : R → R . ◮ Remember, the 2nd derivative test lets you decide if a critical point is a local maximum or minimum if the 2nd derivative is nonzero. We thus expect to be able to describe M in relation to these points. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 12 / 1024

Morse Lemma Theorem (Lemma of Morse) Let f ∈ C ∞ ( M , R ) , and let p ∈ M be a nondegenerate critical point of f . Then there exists a neighborhood U ⊂ M of p and a coordinate system ( y 1 , . . . , y n ) on U such that y i ( p ) = 0 for all 1 ≤ i ≤ n , and moreover f = f ( p ) − ( y 1 ) 2 − · · · − ( y λ ) 2 + ( y λ +1 ) 2 + · · · + ( y n ) 2 where λ = λ ( p ) is the index of p. Corollary If p ∈ M is a nondegenerate critical point of f , then it is isolated. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 13 / 1024

Given f : M → R , define the ‘half-space’ M a = f − 1 ( −∞ , a ] = { x ∈ M : f ( x ) ≤ a } . Theorem (Milnor) 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 furthermore M a is a deformation retract of M b . Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 14 / 1024

The gradient of f induces a local 1-parameter family of diffeomorphisms φ t : M → M away from critical points. Thus allowing the points of M a to flow along these gives the desired deformation retract. Remark: The condition that f − 1 ([ a , b ]) be compact cannot be relaxed. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 15 / 1024

Theorem (Milnor) Let f : M → R be C ∞ and let p ∈ M be a (nondegenerate, isolated) critical point of f . Set c = f ( p ) and λ = λ ( p ) to be the index of p. Suppose there exists ǫ > 0 such that f − 1 ([ c − ǫ, c + ǫ ]) is compact and contains no critical points of f other than p. Then for all sufficiently small ǫ , M c + ǫ has the homotopy type of M c − ǫ with a λ -cell attached. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 16 / 1024

The key observation is that when crossing a critical point, the Morse Lemma is applicable. It can be shown that attaching a λ -cell e λ to M c − ǫ along the ( y 1 , . . . , y λ ) axis, M c − ǫ ∪ e λ ∼ = M c + ǫ . Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 17 / 1024

Intuitively then, the manifold can be constructed from cells determined by the indices of the critical points. Theorem (Milnor) If f : M → R is Morse and for all a ∈ R it holds that M a is compact, then M has the homotopy type of a CW complex with one cell of dimension λ for each critical point with index λ . Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 18 / 1024

This is enough to get a few results. For example, Theorem (Reeb) Let M be a compact smooth manifold, and let f : M → R be Morse. If f has exactly two critical points, then M is homeomorphic to a sphere. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 19 / 1024

Differential Forms Recall the space Ω k ( M ) of differential k -forms over M , and the exterior derivative d : Ω k → Ω k +1 , which gives rise to the deRham co-chain complex 0 → · · · d → Ω k ( M ) d → Ω k +1 ( M ) d − − − → · · · → 0 Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 20 / 1024

Betti Numbers The associated cohomology group is the deRham cohomology group dR ( M ) = ker d : Ω k → Ω k +1 H k im d : Ω k − 1 → Ω k and further we define the k -th Betti number of M , β k = dim H k dR ( M ) . This cohomology encodes topological information about the manifold algebraically, and is the starting point for fields such as Hodge Theory and Index Theory. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 21 / 1024

The Betti numbers are topological invariants. They are related to the classical Euler characteristic χ ( M ) by n � ( − 1) k β k . χ ( M ) = k =0 Which is an explicit expression for the following lemma from Index Theory: Lemma Let D = d + δ be the Dirac operator for the Hodge Laplacian ∆ = D 2 = d δ + δ d. Then χ ( M ) = index( D ) where index( D ) = dim ker( D ) − dim coker( D ) denotes the analytic index. Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 22 / 1024

The Betti numbers can be interpreted directly: β 0 is the number of connected components of M . β 1 is the number of 1-dimensional “holes” (nontrivial loops) in M . β 2 is the number of 2-dimensional “cavities” (nontrivial spheres) in M . and so on. Note, the largest nonzero Betti number in a n -dimensional manifold M is β n . Gianmarco Molino (UConn Sigma Seminar) An Introduction to Morse Theory 27 July, 2017 23 / 1024