Differential Topology, fall 2012

This is the web-site for the course "Differential Topology", which will take place during fall 2012. Here you will find all the practical informations about the course, changes that take place during the year, etc.

**Examination:** Wednesday, January 23rd, 2013, 14:00 - 17:00, Wisk.611

**Final results:** here, please contact me soon if you are interested in the retake examination

**Retake examination:** Thursday, February 28th, 2013, 15:00 - 18:00, Wisk.610

**The rules for passing the course:** There is a midterm examination and a final examination. The former accounts for 30% of the final mark,
the latter for 70%. A final mark above 5 is needed in order to pass the course. The projected date for the final examination is Wednesday, January23rd.

The course provides an introduction to differential topology. I plan to cover the following topics: topologies on function spaces, Sard's Theorem, Thom Transversality Theorem, Whitney's embedding Theorem, tubular neighborhoods, intersection theory and - if time permits - I will give an introduction to cobordisms and immersion theory.

** Literature: **

I will use the following books/papers for the preparation of the classes (from which I plan to follow Hirsch and, if time permits, the papers by Smale more closely):

V. Guillemin, A. Pollack, *Differential Topology*, Prentice Hall 1974

M. Hirsch, *Differential Topology*, Springer 1976, reprint 1997

S. Smale, *A Classification of Immersions of the Two-Sphere*, Trans. A.M.S., Vol. 90, No. 2 (1959), 281 - 290

S. Smale, *The Classification of Immserions of Spheres in Euclidean Spaces*, Ann. of Math., Vol. 69,
No. 2 (1959), 327 - 344

** Prerequisits: **

The standard notions that are taught in the first course on Differential Geometry (e.g. the notion of manifolds, smooth maps, immersions and submersions, tangent vectors, the Lie derivative along vector fields, the flow of a tangent vector, the tangent bundle). Towards the end, basic knowledge of Algebraic Topology (definition and elementary properties of homology, cohomology and homotopy groups, (weak) homotopy equivalences) might be helpful, but I will review the relevant constructions and facts in the lecture.

* Week 37/Lecture 1 (September 13, BBL 308):* lecture notes, exercises (solutions)

At the beginning I gave a short motivation for differential topology. Then basic notions concerning manifolds were reviewed, such as: topological properties (Hausdorffness, 2nd countability), atlases, C^r-manifolds, manifolds with boundary. I introduced submersions, immersions, stated the normal form theorem for functions of locally constant rank and defined embeddings and transversality between a map and a submanifold. In the end I established a preliminary version of Whitney's embedding Theorem, i.e. I showed that every compact C^r manifold (r>1) of dimension n can be embedded into Euclidean space of dimension 2n+1. To obtain this result, one first constructs an embedding into some Euclidean space of dimension q (using compactness and a partition of unity), then one shows that if q > 2n+1 one can find a linear projection whose restriction to M is still an embedding.

* Week 38/Lecture 2 (September 20, BBL 161):* lecture notes, exercises
(solutions)

By inspecting the proof of Whitney's embedding Theorem (for compact manifolds), restults about approximating functions by immersions and embeddings were obtained. Then I defined the weak C^r-topology (also called compact-open C^r-topology) in terms of a subbasis. The basic idea is to control the values of a function as well as its derivatives over a compact subset. In the case of smooth functions, one defines the topology as the initial one for the inclusions into C^r-functions, for all r. I started to define the strong C^r-topology (or Whitney C^r-topology), which can be roughly thought of as allowing for some infinite intersections of memebers of the subbasis of the weak topology.

* Week 39/Lecture 3 (September 27, BBL 308):* lecture notes, exercises (

I recalled the definition of the weak C^r-topology in terms of a subbasis. Then I showed that the weak topology on the set of C^r-maps from some open subset
B of R^m to R^n admits a
subbasis, which can be defined without any reference to charts.
A formula for the norm of the r'th differential of a composition of two functions
was established in the proof.
Then the strong C^r-topology (or Whitney C^r-topology) on C^r-maps from M to N was defined. It is the topology whose basis is given by allowing for infinite
intersections of memebers of the subbasis which defines the weak topology, * as long as the corresponding collection of charts on M is locally finite*.
In the end I showed that for M=B an open subset of R^n and N=R^n, the strong topology also admits a description which does not refers to any charts.

* Week 40/Lecture 4 (October 4, BBL 308):* lecture notes, exercises (solutions)

I started with some pointset topology of C^r-manifolds, first proving Whitney's theorem, which says that any closed subset of a C^r-manifold is the zero-set of some non-negative C^r-function. Immidiate consequences are that (1) any two disjoint closed subsets can be separated by disjoint open subsets and (2) for any member of an open cover one can find a closed subset, such that the resulting collection of closed subsets still covers the whole manifold. Then I established local stability (openess) for immersions, submersions and embeddings of class C^1. In the case of immersions and submersions, the local statements directly lead to the openness of the set of C^r-immersions and submersions with respect to the strong topology. Concerning embeddings, one first ueses the local result to find a neighborhood Y of a given embedding f in the strong topology, such that any map contained in this neighborhood is an embedding when restricted to the memebers of some open cover. One then finds another neighborhood Z of f such that functions in the intersection of Y and Z are forced to be embeddings.

* Week 41/Lecture 5 (October 11, BBL 069):* lecture notes, exercises (solutions)

I finished the proof that C^r-embeddings form an open subset in the strong topology. I then proved that the set of proper maps is open (in the strong C^0-topology). As a consequence, the set of closed C^r-embeddings and the set of C^r-diffeomorphisms are open as well. Homeomorphisms (i.e. C^0-diffeomorphisms) are not open in the strong C^0-topology, but every homeomorphism admits a neighborhood which consists of surjective maps. The proof of this relies on the fact that the identity map of the sphere is not homotopic to a constant map. I outlined a proof of the fact.

* Week 42/Lecture 6 (October 18, BBL 308):* lecture notes, exercises (solutions)

I proved the basic approximation result, which says that if M and N are C^r-manifolds, then the set of C^r-maps is dense in the set of C^s-maps from M to N, when the latter is equipped with the strong C^s-topology (s is a natural number ranging from 0 to r). Consequently, C^r-immersions / submersions / (closed) embeddings / diffeomorphisms are dense inside the set of C^s-immersions / submersions / (close) embeddinge / diffeomorphisms. The proof consists of an inductive procedure and (a relative version of) an apprixmation result for maps between open subsets of Euclidean spaces, which is proved with the help of convolution kernels.

* Week 43/Lecture 7 (October 25, DDW 1.36):* lecture notes, exercises
(solutions for exercise 1 & 2, solution for exercise 3)

I first proved that every C^r-manifold admits a compatible smooth structure. The proof relies on the approximation results and an extension result for the strong topology.

Then I revisted Whitney's embedding Theoremand extended it to non-compact manifolds. I showed how to deduce Whitney's embedding Theorem from the fact that the set of (injective) immersions from M to N is dense in all C^r-maps with respect to the strong topology if 2 dim(M) >= dim(N) (2 dim(M)> dim(N)). This, in turn, was proven by globalizing the corresponding denseness result for maps from a closed ball to Euclidean space.

* Week 44/ No lecture (November 1, DDW 1.36):* examination (results)

* Week 45/Lecture 8 (November 8, WENT OC109):*
lecture notes

I introduced the space of s-jets J^s(M,N) between two C^r-manifolds, equipped them with a topology and a C^(r-s)-structure and showed that they are completely metrizeable. Then I defined the compact-open and strong topology on the set of continuous functions between topological spaces. The C^r-maps from M to N can be seen as a subset of the set of C^0-maps from M to J^r(M,N). This subspace is closed with respect to the weak topology and the topology on C^r(M,N) induced from the compact-open (strong) topology coincides with the weak (strong) C^r-topology that we previously used (these statements are not true in the smooth case!). As a consequence, C^r(M,N) with the weak C^r-topology is completely metrizeable and hence a Baire space. In the end we proved that this is also true for any weakly closed subset of C^r(M,N) with the strong C^r-topology.

* Week 46/Lecture 9 (November 15, WENT OC109):* lecture notes, exercises (solutions)

Subsets of manifolds that are of measure zero were introduced. Then a version of Sard's Theorem was proved. It asserts that the set of all singular values of any smooth manifold is a subset of measure zero. As an immediate consequence, we saw that the k'th homotopy groups \pi_k(S^n) of the n-dimensional sphere are trivial provided k is strictly smaller than n.

In the second part of the lecture, C^r-mapping classes on a pair of manifolds (M,N) were defined. These should be thought of as a formalization of the sheaf of maps from M to N that "satisfy some local condition imposed on a closed subset L of M". A C^r-mapping class is called rich if it assigns to sufficiently small subsets a dense and open subset of C^r-mappings (with respect to the weak topology). The Globalization Theorem says that the openess and denseness still holds globally with respect to the weak topology if the closed subset L is compact and with respect to the strong topology in general.
* Week 47/Lecture 10 (November 22, WENT OC109):* lecture notes, exercises (solutions)

Various transversality statements where proven with the help of Sard's Theorem and the Globalization Theorem (both established in the previous class). The definition of richness of a C^r-mapping class was stated wrongly last time and was fixed. Then I proved the Transversality Theorem (TT), saying that "most" maps from M to N are transversal to some given submanifold A of N. I also proved the parametric version of TT and the jet version. The latter asserts that the r'th jet of "most" maps from M to N is transversal to some given submanifold A of the r'th jet bundle J^r(M,N). As an application of the jet version, I deduced that the set of Morse functions on a smooth manifold forms an open and dense subset with respect to the strong topology.

* Week 48/Lecture 11 (November 29, WENT OC109):* lecture notes, exercises
(solutions for exercise 1 & 2)

I proved homotopy invariance of pull backs. This reduces to proving that any two vector bundles which are concordant (i.e. are isomorphic to the restrictions of a vector bundle defined over M times [0,1] to t=0 and t=1, respectively) are isomorphic. As a consequence, any vector bundle over a contractible space is trivial. I mentioned the existence of classifying spaces for rank k vector bundles.

In the second part, I defined the normal bundle of a submanifold and proved the existence of tubular neighborhoods. As an application, I proved that, given a continuous map f between C^r-manifolds, one can find a neighborhood in the strong C^0-topology all of whose elements are homotopic to f. Hence any continuous map can be approximated by homotopic C^r-maps. In the end I defined isotopies and the vertical derivative and showed that all tubular neighborhoods of a fixed submanifold can be related by isotopies, up to restricting to a neighborhood of the zero section and the action of an automorphism of the normal bundle.

* Week 49/Lecture 12 (December 6, WENT OC109):* lecture notes

I first discussed orientability and orientations of manifolds. I presented three equivalent ways to think about these concepts: in terms of equivalence classes of basis (or local frames), in terms of the orientation bundle and in terms of the orientation cover. I also introduced the notion of a map being orientation preserving / reversing at a point.

Then the degree of a C^1-map f between compact, oriented manifold of equal dimension was introduced. It is given by counting the number of points in the preimage of a regular value, where a point x counts as +1 if the map f is orientation preserving at x, and as -1 otherwise. I proved that this definition does not depend on the chosen regular value and coincides for homotopic maps. This allows to extend the degree to all continuous maps.

* Week 50/Lecture 13 (December 13, WENT OC109):* lecture notes, exercises
(solutions)

I continued to discuss the degree of a map between compact, oriented manifolds of equal dimension. The main aim was to show that homotopy classes of maps from a compact, connected, oriented manifold to the sphere of the same dimension are classified by the degree. I used Tietze's Extension Theorem and the fact that a smooth mapping to a sphere, which is defined on the boundary of a manifolds, extends smoothly to the whole manifold if and only if the degree is zero. I defined the linking number and the Hopf map and described some applications.

* Week 50/Lecture 14 (December 14, BBL 069):*lecture notes

I defined the intersection number of a map and a manifold and the intersection number of two submanifolds. I stated the problem of understanding which vector bundles admit nowhere vanishing sections. The existence of such a section is equivalent to splitting the vector bundle into a trivial line bundle and a vector bundle of lower rank. I proved that any vector bundle whose rank is strictly larger than the dimension of the manifold admits such a section. Moreover, I showed that if the rank equals the dimension, there is always a section that vanishes at exactly one point.

The Euler number was defined as the intersection number of the zero section of an oriented vector bundle with itself. I showed that, in the oriented case and under the assumption that the rank equals the dimension, the Euler number is the only obstruction to the existence of nowhere vanishing sections. As applications, we obtained that any vector field on a sphere of even dimension has a zero and consequently the tangent bundle to S^2n is not trivial.