# What is Differential topology: Definition and 18 Discussions

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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1. ### I Fiber bundle homeomorphism with the fiber

Hi, in the definition of fiber bundle there is a continuous onto map ##\pi## from the total space ##E## into the base space ##B##. Then there are local trivialization maps ##\varphi: \pi^{-1}(U) \rightarrow U \times F## where the open set ##U## in the base space is the trivializing neighborhood...
2. ### I ##SU(2)## homeomorphic with ##\mathbb S^3##

Hi, ##SU(2)## group as topological space is homeomorphic to the 3-sphere ##\mathbb S^3##. Since ##SU(2)## matrices are unitary there is a natural bijection between them and points on ##\mathbb S^3##. In order to define an homeomorphism a topology is needed on both spaces involved. ##\mathbb...
3. ### A How Does the Chain Rule Apply to Pushforwards in Differential Geometry?

prove that if ##g:Y→Z## and ##f:X→Y## are two smooth maps between a smooth manifolds, then a homomorphism that induced are fulfilling :## (g◦f)∗=f∗◦g∗\, :\, H∙(Z)→H∙(X)## I must to prove this by a differential forms, but I do not how I can use them . I began in this way: if f∗ : H(Y)→H(X), g∗...
4. ### A Physical meaning of "exotic smoothness" in (and only in) 4D

I see that this has been discussed before, but the old threads are closed. As Carl Brans and others note, it seems too big a coincidence to ignore. Why is exotic smoothness "good" (in the sense that it permits richer physics or something like that)? Exotic Smoothness and Physics,arXiv "there...
5. ### A Tangent Bundle questions about commutative diagram

I don't know how to create a commutative diagram here so I'd like to refer to Diagram (1) in this Wikipedia article. I need to discuss the application of this diagram to the tangent bundle of a smooth manifold because there are some basic points that are either glossed over or conflict in the...
6. ### References for Self Study in de Rham Cohomology

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...
7. ### MHB Differential Topology Notes - at undergraduate level

I am trying to understand Differential Topology using several textbooks including Lee's book on Smooth Manifolds. I am looking for some good online lecture notes at undergraduate level (especially if they have good diagrams and examples) in order to supplement the texts ... Can anyone help in...
8. ### Differential geometry vs differential topology

in a nutshell, what is the difference between those two fields?
9. ### Recomended differential topology books

Hi, I want to study differential topology by myself, and i am looking for a clear book that emphesizes also the intuitive aspect. I will be grateful to get some recommendations. Thank's Hedi
10. ### Differential Topology: Proving Integral of f*dw=0 and Line Bundle over RPn

Hi I would appreciate any help with these problems. Thanks in Advance! 1.Suppose X is the boundary of a manifold W, W is compact, and f : X → Y is a smooth map. Let w be a closed k-form on Y , where k = dimX. Prove that if f extends to all of W then integral of f* dw = 0. Assume that W...
11. ### What's the difference between differential topology and algebraic topology?

Having some knowledge of differential geometry, I want to self-study topology. Which of the two areas shall I study first? Thanks for answer!
12. ### Do I need ODE and PDE for differential topology?

I am a senior in mathematics studying graduate point-set topoology atm. I am thinking I want to study differential topology in graduate school and maybe apply it to problems in cosmology. Do I need to take more ODE and PDE? I took intro to diff eq- the one that all engineering undergrads take...
13. ### Differential Topology: 1-dimensional manifold

Homework Statement Given S1={(x,y) in R2: x2+y2=1}. Show that S1 is a 1-dimensional manifold. Homework Equations The Attempt at a Solution Let f1:(-1,1)->S1 s.t. f1(x)=(x,(1-x2)1/2). This mapping is a diffeomorphism from (-1,1) onto the top half of the circle S1. I was...
14. ### Mathematica Mathematica and differential topology

Solving equations in differential topology by pen and paper, and in Microsoft word has been tedious and error prone. Can Mathematica help? Mathematica would be required to deal with tensor equations on a pseudo Riemann manifold of four dimensions with complex matrix entries. It should be...
15. ### Differential Topology: Essential Concepts Explained

I have what's certainly a totally "newbie" question, but it's something I've been wondering about.. Suppose we have a simple boundary value problem from electrostatics. For instance, suppose we have a conducting sphere held at some potential, \phi = \phi_0. Because the sphere is conducting...
16. ### What are the basics of differential topology?

I was just wondering if anyone had a decent website explaining some of the basic terminology of differential topology. Specifically, I'm having a bit of trouble understanding charts and atlases and how one defines a smooth manifold in an arbitrary setting (i.e., not necessarily embedded in R^n)
17. ### Exploring Differential Topology: Uncovering Answers to Fundamental Questions

I had to ask myself two simple problems in differential topology: 1) Why is the rank of a diffeomorphism (on a manifold of dimension m) of rank m? 2) Why is a chart on a manifold an embedding? These are actually quite obvious so textbooks don't even bother proving it. So I've...
18. ### Learn Differential Topology: Point-Set, Algebraic, & Calculus on Manifolds

I have just found that topology is very interesting. I just want to know how one studies topology. do they go in the order of Point-set Topology, Algebraic Topology, then Differential Topology? My ultimate goal is to understand Calculus on Manifolds and Morse Theory. Is it possible to jump to...