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I am reading a book on motion planning for mobile robots, I have a really hard time with the mathematics. Some parts it is talking about the topology of the space and manifolds and compactness of the space.... I am too lost. I also have some problems with the math notations used in this book. There is an appendix which goes through topology and manifolds but it's only 3 pages and it really fast.

I need a very basic book or paper about topology, which I can read in about a month. Something very basic and with some real applications.

Any other comments are highly appreciated.

Edit: I think I will need to have a basic idea about:
- Topology
- Manifolds
- Group theory
- Homology

I have found some basic books on topology. Would you take a look at them and let me know which one would be more appropriate for me?
1- Topology for Computing
2- Computational Topology
3- Elementary Applied Topology
4- Topology and Robotics (Contemporary Mathematics), I think this one would be great if I have some basic idea about topology

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  • $\begingroup$ I think it would be more efficient for you to go to the mathematics department and find a topologist who's willing to explain the relevant parts to you. $\endgroup$
    – Raskolnikov
    Sep 26, 2013 at 7:45
  • $\begingroup$ You'll presumably need to know some algebra too (notably group theory), as I imagine the braid groups may play a prominent role in your study. $\endgroup$
    – Dan Rust
    Sep 26, 2013 at 9:43
  • $\begingroup$ What's the robotics book? My suggestion is to get another book, unless there's something very special about the current one. $\endgroup$
    – bubba
    Sep 26, 2013 at 12:08
  • $\begingroup$ @Raskolnikov First I prefer to read a book about topology, and if I had a problem I could find a topologist in my university. $\endgroup$
    – Ali
    Sep 26, 2013 at 17:54
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    $\begingroup$ @Ali I think it really depends on your background. As a pure mathematician, my tastes for a 'good topology book' might differ from those of a engineer. Do you have a suitably strong background in pure mathematics (particularly proof based subjects)? $\endgroup$
    – Dan Rust
    Sep 26, 2013 at 18:18

3 Answers 3

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As usrtt1 suggests, Introduction to Topology: Pure and Applied by Adams and Franzosa is a great book introducing topology and many of its applications, although it stops short of material based on homology. An alternative might be Basener's Topology and Its Applications, based on its table of contents, but I do not have the latter book.

There are many abstract algebra texts out there. One place to pick up some basic group theory would be Fraleigh's A First Course in Abstract Algebra, 7th Edition, as this edition contains a light introduction to homology.

Some good suggestions for further reading can be found on Jeff Erickson's list of references for his Computational Topology course. In addition, once you have a fairly strong background in topology and manifolds, Farber's small book: Invitation to Topological Robotics could be of interest.

Update 1: Michael Robinson's Topological Signal Processing, published in 2014, is an introduction to the use of topological tools and sheaves in signal processing. See also Robinson's website.

Update 2: Nicholas A. Scoville's Discrete Morse Theory, published in 2019, is a very accessible introduction to the titular topic. No group theory required, just some linear algebra.

Edit re the four books mentioned in the question:

Topology for Computing and Computational Topology are well worth reading, but might move a little fast for the beginner. However, once you've picked up some basic topology and group theory, they should be much more accessible.

Robert Ghrist's Elementary Applied Topology is intentionally "a quick tour...to motivate...applied topology," according to the author. Ghrist moves swiftly through a dizzying array of topological concepts and applications. It's wonderful inspiration to dive deeper into this material.

As for Topology and Robotics, I do not have this particular book, but it looks like a collection of papers, probably best approached when you have a fairly strong background. You may wish to check the authors' homepages for preprints.

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There's a really good book on topology in engineering mentioned by the others too:

Introduction to Topology - Pure and Applied, by Colin Adams and Robert Franzosa

The nice thing about this textbook is that it gives a detailed application to the concepts after every chapter:

  1. Introduction
  2. Topological Basis -> Mutation of DNA
  3. Closure, Interior and Boundary -> Data in GIS
  4. Creation of Spaces -> Configuration and Phase Space
  5. Continuity -> Robotics
  6. Metric Spaces
  7. Connectedness -> Automated Guided Vehicles
  8. Compactness
  9. Dynamical Systems -> Chaos in Population Model
  10. Homotopy -> Heartbeat Model
  11. Fixed Point Theorem -> Economics
  12. Embeddings -> Digital Image Processing
  13. Knots -> Chirality in Biochemistry
  14. Graphs -> Boiling Point of Molecules, Electronic Circuits
  15. Manifolds -> Geometry of the Universe

For a more detailed table of contents, see http://math.umaine.edu/~franzosa/TOC.htm.

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I have not read this: http://www.amazon.com/Introduction-Topology-Applied-Colin-Adams/dp/0131848690/ but I heard good things about it!

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