Integrating Differential Forms. and closely follow Guillemin and Pollack’s Differential Topology. 2 1Open in the subspace topology. 3. In the winter of , I decided to write up complete solutions to the starred exercises in. Differential Topology by Guillemin and Pollack. Originally published: Englewood Cliffs, N.J.: Prentice-Hall,

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Readership Undergraduate and graduate students interested in differential topology.

In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. 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. Browse the current eBook Collections price list. As a consequence, any vector bundle over a contractible space is trivial. 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 of this relies on the fact that the identity map of the sphere is not homotopic to a constant map.

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Differential Topology

Email, fax, or send via postal mail to:. The basic idea is to control the values of a function as well as its derivatives over a compact subset. Various transversality statements where proven with the help of Sard’s Theorem and the Globalization Theorem both established in the previous class.


In the end I established a preliminary version of Whitney’s embedding Theorem, i.

I continued to discuss the degree of a map between compact, oriented manifolds of equal dimension. 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. In the end I defined isotopies and the vertical derivative and showed that all tubular neighborhoods pollsck 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.

Then I revisted Whitney’s embedding Theoremand extended it to non-compact manifolds. A final mark above 5 is needed in order to pass the course. I proved homotopy invariance of pull backs.

At the beginning I gave a short motivation for differential topology. Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds.

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.

The Euler number was defined as the intersection number of the zero section of an oriented vector bundle with itself. Then a version of Sard’s Theorem was proved. 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.

I defined the intersection number of a map and a manifold and the intersection number of two submanifolds. I presented three equivalent ways to think about these concepts: I outlined a proof of the fact.


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. I also proved the parametric version of TT and the jet version. I proved that this definition does not depend gyillemin the chosen regular value and coincides for homotopic maps. I defined the linking number and the Hopf map and described some applications.

The rules for passing the course: An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. It asserts pollxck the set of all singular values of any smooth manifold is a subset of measure zero. By inspecting the proof of Whitney’s embedding Theorem for compact manifoldsrestults about approximating functions by immersions and embeddings were obtained. 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.

I first discussed orientability and orientations of manifolds.

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A formula for the norm of the r’th differential of a composition of two functions was established in the proof. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. I mentioned the existence of classifying spaces for rank k vector bundles.