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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To subscribe to the current year of Memoirs of the AMSplease download this required license agreement. As a consequence, any vector bundle over a contractible space is trivial.

Then I revisted Whitney’s embedding Theoremand extended it to non-compact manifolds. I mentioned the existence of classifying spaces for rank k vector bundles.

This, in turn, was proven by globalizing the corresponding denseness result for maps from a closed ball to Euclidean space. This reduces to proving that any two vector bundles which are concordant i.

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. I outlined a proof of the fact.

Differential Topology

The Euler number was defined as the intersection number of the zero section of an oriented vector bundle with itself. 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 book has a wealth of exercises of various types. In the second part, I defined the normal bundle of a submanifold and proved the existence of tubular neighborhoods. Towards the end, basic knowledge of Algebraic Topology definition and elementary properties of homology, cohomology and homotopy diffferential, weak homotopy equivalences might be helpful, but I will review the relevant constructions and facts in the lecture.


The course provides an introduction to differential topology. Email, fax, or send via postal mail to: A final mark above 5 is needed in order to pass the course.

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 defined the intersection number of a map and a manifold and the intersection number of two submanifolds. The main aim was to show that homotopy classes of maps from a compact, connected, oriented manifold to the pollaci of the same dimension are classified by the degree. There is a midterm examination and a final examination. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem.

An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. Then a version of Sard’s Theorem was proved.

Pollack, Differential TopologyPrentice Hall I proved homotopy invariance of pull backs.

For AMS eBook frontlist subscriptions or backfile collection purchases: 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. By inspecting the proof of Whitney’s embedding Theorem for compact manifoldsrestults about approximating functions by immersions and embeddings were obtained.


I first discussed orientability and orientations of manifolds. Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. 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. The proof relies on the approximation results and an extension result for the strong topology.

Then basic notions concerning manifolds were reviewed, such as: Complete and sign the license agreement. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra.

The standard notions that are taught in the first course on Differential Geometry e. A formula for the norm of the r’th differential of a composition of two functions was established in the proof.

differential topology

The rules for passing the course: I also proved the parametric version of TT and the jet version. 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.

It asserts that the set of tppology singular values of any smooth manifold is a subset of measure zero.

I presented three equivalent ways to think about differentiql concepts: Subsets of manifolds that are of measure zero were introduced. Readership Undergraduate and graduate students interested in differential topology.