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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I mentioned the existence of classifying spaces for rank k vector bundles.

Differential Topology

The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. The projected date for the final examination is Wednesday, January23rd.

I proved homotopy invariance of pull backs. I continued to discuss the degree of a map between compact, oriented manifolds of equal dimension. This, in turn, was proven by globalizing the corresponding denseness result for maps from a closed ball gujllemin Euclidean space.

I also proved the parametric version of TT and the jet version. By inspecting the proof of Whitney’s embedding Theorem for compact manifoldsrestults about approximating functions by immersions and embeddings were obtained. 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.

Then a version of Sard’s Theorem was proved. The book is suitable for either an introductory graduate course or an advanced undergraduate course. 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 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.


An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance.

In the years since its first publication, Guillemin and Pollack’s book has become a standard text on the pol,ack. Complete and sign the license agreement. It asserts that the set of all singular values of any smooth manifold is a subset of measure zero.

The Euler toplogy was defined as the intersection number of the zero section of an oriented vector bundle with itself. 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.

Differential Topology – Victor Guillemin, Alan Pollack – Google Books

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. 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.

The book has a wealth of exercises of various types. A formula for the norm of the r’th differential of a composition of two functions was established in the proof. I defined the linking number and the Hopf map and described some applications.

Some are routine explorations of the main material. The proof of this relies on the fact that the identity map of the sphere is not homotopic to a constant map.


Email, fax, poplack send via postal mail to:. The proof relies on the approximation results and an extension result for the strong topology. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. Pollack, Differential TopologyPrentice Guilemin I proved topoology any vector bundle whose rank is strictly larger than the dimension of the manifold admits such a section.

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. 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, topooogy smoothly to the gullemin manifold if and only if the degree is zero. Email, fax, or send via postal mail to: Subsets of manifolds that are of measure zero topologgy introduced.

Moreover, I showed that if the rank equals the dimension, there is always a section that vanishes at exactly one point. In the end I established a preliminary version of Whitney’s embedding Theorem, i. I outlined a proof of the fact. There is a midterm examination and a final examination. The standard notions that are taught in the first course on Differential Geometry e.