Morse Theory

15 dezembro 2011, 09:30 • Gustavo Granja

Proof of Morse's Lemma.

If f:M --> R has no critical points in f^{-1}[a,b] then M is diffeomorphic to a cylinder f^{-1}(a)x[a,b] by a diffeomorphism intertwining f and the projection on the second component.

Corollary(Reeb) A compact manifold without boundary admitting a Morse function with only two critical points is homeomorphic to a sphere.

Remarks about exotic S7s.

Theorem: If f^{-1}[a,b] contains a unique non-degenerate critical point of index k then f^{1}(b) deformation retracts onto f^{-1}(a) union a k-disk whose intersection with f^{-1}(a) is the S^{k-1} sphere.

Effect of crossing a critical point on deRham cohomology.

Intersection numbers, Euler characteristic and intro to Morse Theory

13 dezembro 2011, 13:00 • Gustavo Granja

This was a 3 hour lecture in room 4.35.

If \alpha is an n-form on N then the integral of f^*\alpha over M is deg(f) times the integral of \alpha over N.

Theorem: M compact, of dimension n, without boundary. Homotopy classes of maps from M to S^n are classified by their degree.

Remarks on what happens if M is not oriented or has boundary.

Definition of intersection number and Euler number of a vector bundle. Examples. If \xi has a non-vanishing section then the Euler number vanishes.

Computation of the Euler characteristic of a vector bundle in terms of the index of a section which is transverse to the zero section.

Examples. Relation of the Euler number to the Euler characteristic calculated from a triangulation. If M is odd dimensional \xi(M)=0. The Euler number of S^n is 1+(-1)^n.

Corollary (Hairy ball theorem): S^{2n} does not admit a nowhere vanishing vector field.

Statement of the Theorem that if M is compact without boundary and \xi(M)=0, then M admits a nowhere vanishing vector field.

Definition of Morse function as a function whose jet is transverse to MxRx0 inside J^1(M,R).

Corollary: Morse functions form an open residual set inside smooth functions.

Definition of the Hessian of a function at a critical point. Statement of Morse's Lemma.

The canonical example of a Morse function: the height function on the torus.

The degree of a map

12 dezembro 2011, 09:00 • Gustavo Granja

Correction of some mistakes in the proof of Thom's theorem from last time.

Definition of degree of a map between smooth manifolds which are oriented and without boundary.

Theorem: The degree is independent of the choice of regular value and depends only on the homotopy class of the map in question.

Some basic examples.

Thom's Theorem

6 dezembro 2011, 14:00 • Gustavo Granja

Converse to the Pontryagin-Thom construction. Given a continuous map \alpha:S^{n+k} --> Th(\gamma^k_{n+k}) make \alpha smooth and transverse to the 0-section, the inverse image of the 0-section is a manifold of dimension n in S^{n+k} well defined up to cobordism.

The effect of composing an embedding of M in \R^{n+k} with the inclusion of \R^{n+k+1} on the Pontryagin-Thom construction is to suspend the original map and then compose it with the canonical map \Sigma Th(\gamma^k_{n+k}) --> Th(\gamma^{k+1}_{n+k+1}).

Definition of n-th homotopy group of a pointed space. This is an abelian group with respect to an operation defined as in the case n=1.

Definition of direct limit of a sequence of abelian groups and homomorphisms.

Def: \pi_n MO = \lim_{k-->\infty} \pi_{n+k} Th(\gamma^k_{n+k} where the homomorphisms are given by suspension, followed by composition with the canonical map.

Let \mathfrak N_n be the unoriented cobordism group of n dimensional manifolds.

Theorem(Thom) The maps \phi: \mathfrak N_n \to \pi_n MO and \psi: \pi_n MO \to \mathfrak N_n where \phi is defined by applying the PT construction to any embedding of a representative and \psi is defined by taking a representative which is smooth and transverse to the 0-section and considering the cobordism class of the inverse image of the 0-section, are inverse isomorphisms.

Proof of Thom's theorem.

The Pontryagin-Thom construction

6 dezembro 2011, 12:30 • Gustavo Granja

Lemma: M compact without boundary, A,X without boundary, and A closed submanifold of X. If H:M x [0,1] --> X is a smooth homotopy between f,g and H,f,g are transverse to A, then H^{-1}(A) is a cobordism between f^{-1}(A) and g^{-1}(A).

Definition of Thom space of a vector bundle. It is homeomorphic to D(E)/S(E), the quotient of the disk bundle by the sphere bundle for any choice of metric on the bundle.

The Thom space is functorial with respect to maps of vector bundles.

Examples: Thom space of a trivial bundle. The Thom space of the tautological line bundle over RP^n is RP^{n+1}. Th(\xi \oplus \R) is the reduced suspension of Th(\xi), \Sigma Th(\xi).

Applying the last example to the tautological bundles over the Grassmannians we get canonical maps

\Sigma Th(\gamma_n^k) --> Th(\gamma^{k+1}_{n+1}).

The Pontryagin-Thom construction on the embedding of a compact manifold without boundary M of dimension k in R^n, producing a continuous map S^n --> Th(\gamma_n^k).