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 course provides an introduction to differential topology. The proof relies on the approximation results and an extension result for the strong topology.
There is a midterm examination and a final examination. Then I defined the compact-open and strong topology on the set of continuous functions between topological spaces. Complete and sign the license agreement. 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.
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. It asserts that the set of all singular values of any smooth manifold is a subset of measure zero. Browse the current eBook Collections price list. Pollack, Differential TopologyPrentice Hall It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within.
In the end I established a preliminary version of Whitney’s embedding Theorem, i. The book is suitable for either an introductory graduate course or an advanced undergraduate course. 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. At the beginning I gave a short motivation for differential topology. I stated the problem of understanding which vector bundles admit nowhere vanishing sections.
A final mark above 5 is needed in order to pass the course. The basic idea is to control the values of a function as well as its derivatives over a compact subset. I outlined a proof of the fact. 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.
Readership Undergraduate and graduate students interested in differential topology. 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.
Subsets of manifolds that are of measure zero were introduced. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. This, in turn, was proven by globalizing the corresponding denseness result for maps from a closed ball to Euclidean space. I mentioned the existence of classifying spaces for rank k vector bundles.
One then finds another neighborhood Z of f such that functions in the intersection of Y and Z are forced to be embeddings. 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.
Moreover, I showed that if the rank equals the dimension, there is always a section that vanishes at exactly one point. Email, fax, or send via postal mail to:. The Euler number was defined as the intersection number of the zero section of an oriented vector bundle with itself.
Some are routine explorations of the main material. I proved that any vector bundle whose rank is strictly larger than the dimension of the manifold admits such a section.
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.
This allows to extend the degree to all continuous maps. Various transversality statements where proven difgerential the help of Sard’s Theorem and the Globalization Theorem both established in the previous class.
I plan to cover the following topics: Polkack proof of this relies on the fact that the identity map of the sphere is not homotopic to a constant map. I continued to discuss the degree of a map between compact, oriented manifolds of equal dimension. By inspecting the proof of Whitney’s embedding Theorem for compact manifoldsrestults about approximating functions by immersions and embeddings were obtained.
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Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. For AMS eBook frontlist subscriptions or backfile collection purchases: I proved that this definition does not depend on the chosen regular value and coincides for homotopic maps.
Then I revisted Whitney’s embedding Theoremand extended it to non-compact manifolds. The book has a wealth of exercises of various guillemn. 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. I proved homotopy invariance of pull backs. By relying on a unifying idea—transversality—the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results.
I first discussed orientability and orientations of manifolds.