Lecture summaries:

15/09/08: Manifolds and differential structures. Differentiable maps. The tangent bundle.
17/09/08: Submanifolds, embeddings and immersions. A submanifold is the same as the image of an embedding. Critical and regular points and values. The inverse image of a regular value is a submanifold. Any manifold can be embedded in R^N for N big.
19/09/08: Maps whose derivative has constant rank are locally diffeomorphic to linear projections. Definition of a map being transverse to a submanifold. The inverse image of a submanifold by a transverse map is a submanifold. Proof of the easy Whitney embedding theorem: Any C^2 map f:M --> R^n can be approximated by an embedding if n>2 dim M and by an immersion if n>2 dim M -1.
24/09/08: Manifolds with boundary. Definition of submanifold and neat submanifold. If f:M --> N and its restriction to the boundary are transverse to a neat submanifold A or the boundary of A is empty, then f^{-1}(A) is a neat submanifold.
26/09/08: Topologies on the sets C^r(M,N): the weak topology and the Whitney or strong topology. Detailed discussion of the example when M is the natural numbers and N is the real numbers. Proof that immersions and submersions are open subsets in the Whitney topology.
1/10/08: Proof that embeddings form an open set in the Whitney topology. Equivalent characterizations of proper maps between manifolds: 1. f^{-1}(K) compact for K compact; 2. x_n a sequence with f(x_n) convergent has a convergent subsequence; 3. f is closed and the fibers f^{-1}(y) are compact.
3/10/08: Proper maps, closed embeddings and diffeomorphisms are open in the Whitney topology. Convolution kernels. Uniform approximation by smooth functions on compact subsets. Smooth functions are dense in C^r(U,V) for U,V open subsets of R^n and R^m. Relative version of the same statement.
8/10/08: Let M and N be manifolds without boundary. C^s(M,N) is dense in C^r(M,N) for s>r. For s>r>=1, if M and N are C^s manifolds which are C^r diffeomorphic then they are C^s diffeomorphic. For s>r>=1, every C^r differential structure contains a C^s differential structure which is unique up to diffeomorphism.
10/10/08: Approximation by smooth functions on open sets of half planes including approximation relative to the boundary. The results from the previous lecture about manifolds without boundary now go through as before for manifolds with boundary. Structure functors. Abstract method for globalizing structures.
15/10/08: For M and N C^r manifolds with r>=1 and dim N >= 2 dim M, C^r immersions are dense in C^r(M,N). Every paracompact manifold has a complete metric. A countable product of complete metric spaces has a complete metric.
17/10/08: For X paracompact and Y complete metric C(X,Y) with the weak topology has a complete metric and any weakly closed subset is a Baire space in the strong topology. Jets. J^r(M,N) is a C^0 manifold for r finite.
22/10/08: The prolongation map is a homeomorphism from C^r(M,N) to a weakly closed subset of C^0(M,J^r(M,N)) in the weak and strong topologies. For M and N C^r manifolds with 1<=r<=infty, C^r(M,N) has a complete metric for the weak topology and is a Baire space in the Whitney topology. If dim N>= 2 dim M +1, embeddings are dense in proper maps in the strong topology. Statement of Sard's theorem. C^1 maps take sets of measure 0 to sets of measure 0.
24/10/08: Proof of Sard' Theorem in the smooth case. Hirsch's proof of Brouwer's fixed point theorem. Statement of transversality theorem (transverse maps form a residual set). Application: The Stiefel manifold V(k,n) of k frames in R^n is (n-k-1)-connected.
29/10/08: C^r mapping functors. Globalization theorem for rich mapping functors. Proof that the C^r mapping functor X_L(U,V)= C^r maps from U to V transverse to A at L is rich when N is without boundary. Statement of the transversality theorem: transverse maps are residual.
31/10/08: Conclusion of the proof of the transversality theorem. Parametric transversality. Example: Any embedding is regularly homotopic to an embedding transverse to a given submanifold. Statement of the jet transversality theorem.
5/11/08: Proof of the jet transversality theorem. Application: a trivial proof that immersion of $M$ in $N$ are dense in C^2_S(M,N) as long as dim N>= 2dim M.
7/11/08: Bundles with fiber F and structure group G. Examples. Classification of bundles in terms of transition functions. Constructions induced by homomorphisms of the structure group.
12/11/08: Constructions with bundles: pullback and reduction of structure group. Reductions of structure group from G to H amount to sections of the associated bundle with fiber G/H. Every vector bundle over a paracompact base has a metric. The Whitney sum. The quotient by a subbundle and the kernel of a epimorphism are bundles. If B is paracompact all s.e.s sequences of vector bundles over B split. Statement of the covering homotopy theorem for maps of bundles and consequences: bundles over cylinders are products and homotopy invariance of pullbacks.
14/11/08: Proof of the homotopy covering theorem for bundle maps. If M is a smooth manifold any vector bundle embeds in a trivial bundle over M. Correspondence between such maps and pullbacks of the tautological bundle over the Grassmanian. Classification of vector bundles over manifolds.
19/11/08: Orientations of vector bundles as sections of an orientation two-fold cover of the base. Every vector bundle over a simply connected base is orientable. Orientable double coverings of manifolds. S^2n is the orientable double covering of RP^2n. A rank 1 bundle over a paracompact base is orientable iff it is trivial. The normal bundle of the boundary of a manifold is trivial. Separation properties of codimension 1 closed submanifolds.
21/11/08: A closed connected codimension 1 submanifold M of a connected boundaryless manifold N disconects it. Tubular neighborhoods and collars. Every boundary admits a collaring and every neat submanifold admits a tubular neighborhood.
26/11/08: Definition of isotopy. Isotopy is an equivalence relation. Isotopies of tubular neighborhoods. Collars and tubular neighborhoods are unique up to isotopy. Definition of degree and degree mod 2 of a map between compact manifolds of the same dimension without boundary.
28/11/08: The degree of a map is well defined and invariant under homotopy. A map of nonzero degree is surjective. Every tangent vector field to an even sphere vanishes at some point. The Hopf theorem classifying homotopy classes of maps from a manifold of dimension n to S^n.
03/12/08: Intersection numbers of maps with submanifolds. The euler number of an oriented vector bundle of rank n over a compact oriented manifold of dimension n. The Euler characteristic. The index of a vector field at a point. Characterization in terms of maps between spheres. How to compute the Euler characteristic. Examples.
05/12/08: The Euler characteristic of a product. The Euler number of an oriented bundle over an odd dimensional manifold vanishes. Existence of non-vanishing vector fields over manifolds with boundary. The Euler number vanishes if and only if there is a non-vanishing section.
10/12/08: Cobordism and oriented cobordism. The Pontryagin-Thom construction. Thom's theorem identifying the cobordism groups with homotopy groups of the Thom space of the universal bundle.
12/12/08: Non-degenerate critical points. Morse functions. The set of Morse functions is open and dense in the strong topology. Proof of Morse's Lemma giving a normal form for a Morse function near a critical point.
17/12/08: Review of existence, uniqueness and dependence on initial conditions for solutions of autonomous odes. Flows on manifolds. If f:M-->[a,b] is a function with no critical points with M compact and f(boundary of M)={a,b} then M is diffeomorphic to f^{-1}(a)x[a,b]. Reeb's Theorem: A compact manifold without boundary which admits a Morse function with only two critical points is homeomorphic to a sphere.
19/12/08: The Ehreshmann fibration theorem. If f(M)=[a,b], f(bd of M)={a,b} with a and b regular values and M contains a unique critical point of index k then f^{-1}(b) deformation retract onto f^{-1}(a) with an embedded k-cell attached. Every compact manifold has the homotopy type of a finite CW complex. The Morse inequalities.