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Need simple introduction to homology / cohomology theory

  1. Feb 24, 2010 #1
    I'd appreciate it if someone points me to a simple and gentle introduction to homology / cohomology theory. By simple and gentle, I mean start out by drawing a triangle and a circle and then connect the two to chains, and homology groups.
     
  2. jcsd
  3. Feb 24, 2010 #2

    Chris Hillman

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    Well, there are certainly books which keep things very simple, e.g. Chapter 14 of Bamberg and Sternberg, A Course in Mathematics for Students of Physics, Vol 2, Cambridge University Press, gives a very simple introduction to simplicial homology. Chapter 13 of Frankel, Geometry of Physics, gives an overview of de Rham cohomology (see also the classic by Flanders, Applications of Differential Forms to the Physical Sciences, Dover reprint). Hatcher, Algebraic Topology is gorgeously illustrated and by far the easiest introduction to cellular homology. But I don't think any of these (or others I could name, like Greenberg & Harper) really fits your bill.

    I should say that most graduate level algebraic topology courses grapple with three versions of homology and probably several versions of cohomology. You need
    • simplicial homology/cohomology (easiest definition of boundary map, cup product) to get started
    • singular homology (powerful but abstract) for efficiently proving crucial theorems,
    • cellular homology for efficient applications to interesting examples,
    • de Rham cohomology for important stuff on compact Lie groups.

    I don't know why there is not (as far as I know) a book on "Algebraic Topology for Working Scientists", which aims to quickly get adult learners up to speed in actually using this stuff. My goal in the still developing thread "BRS: Cellular Homology with Macaulay2" has been to supply a tutorial on using some computer tools which would be accessible to people who have already studied algebraic topology, but as it happens, I've been mulling a "BRS: Simplicial Homology with Macaulay2" which would be more elementary.

    To keep my expository tasks manageable, what about combining Hatcher's book with my tutorials so you can play on your computer? I forgot to give the link to his home page
    http://www.math.cornell.edu/~hatcher/
    where you can download of a free ecopy of several of his textbooks, including Algebraic Topology--- although I highly encourage everyone to buy a copy of the printed version of the latter book, which is published by my favorite mathsci publisher, Cambridge University Press. Hmm... since I more or less confessed to drawing a few of the pictures in that book when I was an undergraduate (Hatcher started writing it long before it was finally published), I should say that I have no financial stake in CUP or that book.

    I think I can say just enough in the planned BRS: Simplicial Homology with Macaulay2 to get you started, especially if you are willing to look into Hatcher's book too.

    [EDIT: Overnight, I thought better of this plan, and was going to say so, but Haelfix beat me to it.]
     
    Last edited: Feb 25, 2010
  4. Feb 25, 2010 #3

    Haelfix

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    Hatcher is pretty brutal for a first time student. Typically its given to graduate students in the mathematics department, and supplemented with lectures and study groups. Or at least, that was how it went for me in grad school. A lot of the homework questions are proofs and designed for mathematicians.

    Its a real tough course for a physics major (senior or 1st year grad student), and it really helps if you are first exposed to the material gently in physics first (often done in a 'geometry of physics' class). I sort of like Nakahara for that, but even that is cursory at best and not necessarily super transparent for a first timer. There really isn't a super easy way to learn the material that exists to my knowledge, outside of lecture notes.
     
  5. Feb 25, 2010 #4

    Chris Hillman

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    I can't disagree. It's a perfect book for those who have studied algebraic topology previously, but probably not suitable for a first textbook.

    I really don't know of any book which does a passable job of introducing algebraic topology for students in non-math fields. There's no denying a good deal of background is required (e.g. abelian groups, rings, R-modules, commutative algebra, homological algebra) but I think that if the author is satisfied with a less than rigorous development and if the readers are willing to accept to accept citations to other texts for full proofs, it should be possible for such a book to exist. I do think Hatcher is correct to stress cellular homology more than many other authors though--- because this style of homology provides the best blend of power, applicability, and convenience.

    I plan to let percolate my plan to write a "BRS: Simplicial Homology with Macaulay2" for a few days, which I hope will improve the readability of the final result.
     
    Last edited: Feb 25, 2010
  6. Feb 25, 2010 #5

    blechman

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    Really?

    I took Algebraic Topology (grad course) as a senior undergraduate using Greenberg-Harper (now THAT is a brutal book!!!), and I retook the course again as a grad student (in physics), this time using Hatcher. I never learned this stuff in a physics context, at least in the classroom.

    Maybe it was my previous experience, but I thought Hatcher was AMAZINGLY clear! I would certainly recommend it.

    I also used Massey's "Algebraic Topology: an Introduction" as a self-taught summer prereq for taking the grad course as an undergrad (assigned by the prof). It was great, but it's mostly fundamental-group stuff, maybe not what you want.
     
  7. Feb 25, 2010 #6

    Chris Hillman

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    Massey's textbook (which I love; it was my undergraduate homotopy book too) only covers homotopy theory (fundamental group, covering spaces). Reading the appendix on homogeneous spaces was a seminal event for me (see if you can spot the connection it suggests between Boltzmann entropy, i.e. log of a multinomial coefficient, and a certain action by a certain group).

    Just to be clear: I also think that Hatcher's book is very clear, full of valuable insight, and very readable. Possibly best enjoyed by "returning students" who have previously been exposed to other algebraic topology textbooks, and will appreciate a labor of love. But these remarks apply to math students, probably not to physics students who, for example, may not have first studied "modern algebra" (groups, rings, modules).

    I am happy, incidently, to find reassurance that algebraic topology hath its fans at PF! I guess we'd all like to see this fun and useful stuff become better known.
     
  8. Feb 25, 2010 #7

    blechman

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    Neat! I'll have to dig it out again.

    i don't think of myself as a "math student", but I guess I was... (am?!....)

    Yeah, this stuff is very neat. The problem for me is that I never use it in my research, and therefore anything I learn I almost immediately forget! :frown:
     
  9. Mar 3, 2010 #8

    Chris Hillman

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    twofish (and anyone else interested in this), "BRS: Simplicial Homology with Macaulay2"
    https://www.physicsforums.com/showthread.php?t=382725
    is my best attempt to motivate the boundary map (actually, it is kind of amazing that none of the half dozen textbooks I looked at attempt to do this!) and to provide an overview of homology theory generally. I plan to follow up by presenting simplicial complexes for the real projective plane and some other favorite surfaces, and to compare their reduced homologies with their homologies as computed in "BRS: Cellular Homology with Macaulay2".
     
  10. Jan 2, 2011 #9

    mathwonk

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    have you looked at various books with the word "combinatorial" in the title?

    such as: A Combinatorial Introduction to Topology [Paperback]
    Michael Henle (Author

    or the books by aleksandrov, et al...
     
  11. Jan 2, 2011 #10

    mathwonk

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    another suggestion is to ask yourself what sort of curves you can prove the cauchy integral theorem for. basically the cauchy theorem says that if a curve (or cycle) bounds something in the domain of holomorphicity, then the integral is zero. this becomes the statement that holomorphic differentials have zero integral over cycles which are zero in homology. so homology classes are the equivalence classes of "curves" that holomorphic integrals are constant on.
     
  12. Jan 2, 2011 #11

    Chris Hillman

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    That is a very important point--- I hope I remembered to mention it in at least one of the other BRS threads on homology.
     
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