References for Self Study in de Rham Cohomology

In summary, the person is self-studying de Rham cohomology from Guillemin and Pollack's book, which presents results as exercises with hints. They have encountered difficulties and are looking for alternative references that use similar definitions. They receive a suggestion to check out chapters 7-8 of Spivak's Differential Geometry Vol. 1 or chapters 4-5 of Singer and Thorpe's Lecture Notes on Elementary Topology and Geometry.
  • #1
MissMoneypenny
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I've been trying to self study the section on de Rham cohomology in Guillemin and Pollack's book Differential Topology. The section is in a sense hands on: most of the results are presented as exercises scattered throughout the section, and some hints are given. I've hit a road block in a few of the exercises and have been searching for a book, some online course notes or some other reference on de Rham cohomology to help me through the exercises I'm stuck on. However, Guillemin and Pollack use slightly different definitions than all of the other books or notes I've been able to find. I'd like to find a book, set of notes, or some other reference that uses similar definitions to Guillemin and Pollack. Can anyone who is familiar with Guillemin and Pollack's book point me in the direction of an alternate reference that treats de Rham cohomology in a similar manner to GP? Thank!
 
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  • #2
try chapters 7-8 of spivak's differential geometry vol. 1, or chapters 4-5 of singer and thorpe's lecture notes on elementary topology and geometry.
 
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  • #3
mathwonk said:
try chapters 7-8 of spivak's differential geometry vol. 1, or chapters 4-5 of singer and thorpe's lecture notes on elementary topology and geometry.
Thanks a lot for your suggestions. I'll head to the library and have a look at those books.
 

1. What is de Rham cohomology?

De Rham cohomology is a mathematical concept that is used to study the topological properties of smooth manifolds. It is a tool that allows for the classification and measurement of holes in a manifold, providing a way to understand its underlying geometric structure.

2. Why is de Rham cohomology important?

De Rham cohomology is important because it provides a powerful tool for understanding the topological properties of smooth manifolds. It allows for the classification of manifolds and can provide insight into their geometric structure, making it a useful tool in many areas of mathematics and physics.

3. How can I learn about de Rham cohomology?

There are many resources available for learning about de Rham cohomology. Some recommended references for self-study are "Introduction to Smooth Manifolds" by John M. Lee, "Differential Forms in Algebraic Topology" by Raoul Bott and Loring W. Tu, and "Algebraic Topology" by Allen Hatcher.

4. What background knowledge do I need to understand de Rham cohomology?

To understand de Rham cohomology, it is helpful to have a strong foundation in multivariable calculus and linear algebra. Some knowledge of point-set topology and abstract algebra may also be useful, but not necessary.

5. How is de Rham cohomology used in real-world applications?

De Rham cohomology has a wide range of applications in mathematics and physics. It is used in the study of smooth manifolds, differential equations, and algebraic topology. It also has applications in areas such as fluid dynamics, electromagnetism, and general relativity.

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