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Quotient Topology and Adjunction Space

  1. Sep 28, 2012 #1
    Does anyone have any good reference to exercises concerning these topics? I would like to understand them better. Thank you.
     
  2. jcsd
  3. Sep 28, 2012 #2

    mathwonk

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    this isa fundamental construction in topology, the building up of fairly general spaces, like manifolds, from conjoining pieces which are simpler, like cells. Thus any discussion of cellular spaces, cellular homology, would discuss this, e.g. I presume the well known free algebraic topology book by A. Hatcher.

    http://www.math.cornell.edu/~hatcher/AT/ATchapters.html


    the appendix has a lot of properties of CW complexes, an important type of quotient or adjunction space.

    if you want more elementary topics on those, a basic topology book might help, like kelley or maybe munkres?

    here are some more free ones.

    http://www.math.jhu.edu/~jmb/note/pushout.pdf

    http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

    here is maybe ,ore what you want: in introduction to topological manifolds by john lee:

    pages 73-86? the problems start on page 81.

    http://books.google.com/books?id=Me...epage&q=adjunction spaces in topology&f=false


    by the way the point of the "push out" nonsense, is just that it is easy to define continuous maps OUT OF an adjunction space. I.e. if you have two maps, one on each piece X and B, and they agree where the two pieces are glued together i.e. along A in B, then they define one map out of the adjunction space (B joined to X along A).
     
    Last edited: Sep 28, 2012
  4. Sep 28, 2012 #3
    Great thanks! I don't much about cells yet, but it seems worthwhile to work ahead. The class will cover some algebraic topology, it just hasn't started yet.

    I think my confusion stems from the fact that our professor didn't really explain why endowing a space (partitioned into subsets of X) with the quotient topology was particularly important, or I was too slow to get it. It seemed that even in defining adjunction spaces, it didn't come much into play except when he simply stated that we take the disjoint union of the two spaces we're trying to glue. I figured it had to do with the fact that the quotient topology made the quotient spaces homeomorphic to other spaces, and in fact, Munkres has a theorem about how a map g:X->Z and the set X*={inverses of the singletons in g} with the quotient topology induces a map f:X*->Z which is homeomorphic iff g is a quotient map. I'm not really sure though.

    Is this the reason, or is there something more subtle I am missing?
     
    Last edited: Sep 28, 2012
  5. Oct 2, 2012 #4

    mathwonk

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    the point is to understand complicated spaces in terms of simpler ones.

    i.e. an interval is simpler than a circle but a circle i q quotient o an interval.
    '
    'a torus is moire complicated than a rectangle but a torus is a quotient of a rectangle,....

    almost any space is a successive union of quotients of rectangles of various dimensions.
     
  6. Oct 3, 2012 #5
    I understand that, I'm just a bit unsure of why the quotient topology is the "right" one. Is it just because then these spaces are homeomorphic to each other (like you said, identifying the endpoints of an interval in the quotient space makes it homeomorphic to the circle in the quotient topology)? Or is there something deeper and more subtle I am missing.

    Like, when I was learning about box vs. product topology in arbitrary products, there were all nice properties about the product topology that the box topology didn't have.
     
  7. Oct 5, 2012 #6

    mathwonk

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    i don't quite understand what is bothering you about the quotient topology. to see why it is what it is, think about how to define the topology on the quotient space consisting of the unit interval with endpoints identified, and then the rectangle with opposite sides identified. and then maybe the sphere with antipodal points identified.

    as i said before (push out), the idea is to define a topology so that a function on the numerator space that is constant on equivalent points, is continuous on the quotient space. think about it.
     
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