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Atrium > Mathematics, Physics & Philosophy: Homotopy theory with finite topological spaces

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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeJul 2nd 2012

    OK, how come nobody told me about this? https://en.wikipedia.org/wiki/Pseudocircle

    As explained in the last sentence, every the geometric realisation of any finite simplicial complex is weakly homotopy equivalent to some finite topological space.

    • CommentRowNumber2.
    • CommentAuthorMatanP
    • CommentTimeJul 3rd 2012

    I had the same feeling about this when I first heard it. I believe the problem is that you cannot model the entire mapping space using fixed finite spaces. every map admits a sufficient subdivision that captures it but all together you’d have to have infinite spaces involved. That said, I think the theory of finite spaces is under-appreciated and should have interesting uses in applied algebraic topology.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJul 3rd 2012

    I once had a thought of doing directed homotopy theory (∞\infty-categories instead of ∞\infty-groupoids) using non-T 1T_1 spaces (with Sierpiński space as the simplest nontrivial example), but it didn’t go anywhere.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 3rd 2012

    Perhaps using pro- and/or ind-finite spaces would help with the problem of mapping spaces?

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeJul 3rd 2012

    Peter May wrote some nice slides and some notes. If you want more look at Jonathan Barmak’s book.

    Raphael Sorkin used (finite) T_0-space in some of his work and I picked up some related ideas in joint work with Gratus and, being me, did explore some of the pro-stuff. (Of course T_0-spaces are just posets, so I think Toby did know about them!)

    • CommentRowNumber6.
    • CommentAuthorjim_stasheff
    • CommentTimeJul 3rd 2012
    Speaking of posets, there is also the realization of the poset as a simplicial complex and then the work of Gerstenhaber relating that (co)homology to Hochschild's.
    • CommentRowNumber7.
    • CommentAuthorjim_stasheff
    • CommentTimeJul 4th 2012
    Following up on that thought:
    if you want to read off homology from a cover, the spaces allowed in the cover should have trivial homology themselves. In this case that means just {a}, {b}, {abc}, {abd}. If you view just these four as a poset, the geometric realization of the nerve is indeed a circle.
    I admit this is pretty much ad hoc, but it also shows how you can construct many more "pseudo" objects.
    Murray

    Is this transverse to what others had in mind?

    jim
    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeJul 5th 2012

    Richard Williamson has some new results in this area and has promised that when he has time he will add some material on them to the lab. They relate to ‘Folk’ model structures and he also mentions the paper: George Raptis, Homology, Homotopy and Applications, vol. 12 (2), 2010, 211-230.

    • CommentRowNumber9.
    • CommentAuthorTim_Porter
    • CommentTimeOct 19th 2018

    Copied from another thread: on Tables of homotopy groups.

    The paper on minimal finite spaces by Barmak and Minian:

    • Jonathan Barmak and G. Minian, Minimal finite models J. Homotopy Relat. Struct. 2 (2007), 127-140. (ArXiv)

    may be of interest.

  10. Funny to see #8, I had completely forgotten about it, but now distantly remember an email conversation in which we discussed this! I should definitely try to put this material up some time!

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