In the winter of , I decided to write up complete solutions to the starred exercises in. Differential Topology by Guillemin and Pollack. 1 Smooth manifolds and Topological manifolds. 3. Smooth . Gardiner and closely follow Guillemin and Pollack’s Differential Topology. 2. Guillemin, Pollack – Differential Topology (s) – Download as PDF File .pdf), Text File .txt) or view presentation slides online.
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The rules for passing the course: There is a midterm examination and a final examination. 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: Pollack, Differential TopologyPrentice Hall The standard notions that are taught in the first course on Differential Geometry e. 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.
At the beginning I gave a short motivation for differential topology. Then basic notions concerning manifolds were reviewed, such as: I introduced submersions, immersions, stated the normal form theorem for functions of locally constant rank and defined embeddings and polack between a map and a submanifold.
In the end I established a preliminary version of Whitney’s embedding Theorem, i. By inspecting the proof of Whitney’s embedding Theorem for compact manifoldsrestults about approximating functions guilleminn immersions and embeddings were obtained. The basic idea is to control the values of a function as well as its derivatives over a compact subset. Diferential formula for the norm of the r’th differential of a composition of two functions was established in the proof.
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. 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.
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.
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.
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. 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. This, in turn, was proven by globalizing the corresponding denseness result for maps from a closed ball to Euclidean space.
Then I defined the compact-open and strong topology on the set of continuous functions between topological spaces.
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.
Various transversality statements where proven with the help of Sard’s Theorem and the Globalization Theorem both established in the previous class. I also proved the parametric version of TT and the jet version. 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 topokogy.
I proved homotopy invariance of pull backs.
This reduces to proving topollgy any two vector bundles which are concordant i. 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. 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.
I first discussed orientability and orientations of manifolds. I presented three equivalent ways to think about these concepts: 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. 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. I defined the intersection number of a map and a manifold and the intersection number of two submanifolds.
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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 guiillemin of nowhere vanishing sections.