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Recently, I was asked to calculate the fundamental group of the space $X= \{a,b,c,d\}$ with open sets generated by $\{ a, c, abc, acd \}$.

Turns out, $\pi_1(X)\cong \mathbb Z$ and in fact, $X$ is the quotient of $S^1$ (with the northern and southern hemispheres identified). But the result was not so easy to prove and this motivates the questions:

  • Is the fundamental group of every connected manifold (other restrictions / generalizations on the manifold are welcome) the fundamental group of a finite space? (Of course, it would not be Hausdorff). (I observe that there are many redundant points on a manifold where homotopy-equivalent loops need not traverse)

  • Is calculating $\pi_1$ of such finite spaces easier than for the given space? (If yes, this gives a method to calculate fundamental groups of many familiar spaces)

Perhaps the fact that -- $\pi_1$ of any CW complex just depends on its 1-skeleton [edit:2 skeleton]-- may be helpful.

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    $\begingroup$ You mean the 2-skeleton, right? $\endgroup$
    – Ryan Reich
    Commented Nov 10, 2010 at 13:22
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    $\begingroup$ I never paused to consider whether $\pi_1(X)$ could be nonzero for non-Hausdorff finite spaces. Is there a simple way of describing your space $X$ as a quotient of $S^1$? You mentioned identifying the two hemispheres in some way!? $\endgroup$
    – Jim Conant
    Commented Nov 10, 2010 at 15:05
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    $\begingroup$ It looks like he means that a is the northern hemisphere (collapsed to a single point), c is the southern hemisphere (likewise collapsed), and b and d are 1 and -1. Then the four sets he mentioned are the two open hemispheres and the complements of the two other points, which are indeed (with $a \cup b$ and $\emptyset$) the saturated open sets for the quotient of $S^1$ collapsing each hemisphere to different points. $\endgroup$
    – Ryan Reich
    Commented Nov 10, 2010 at 15:58
  • $\begingroup$ Sorry I had to be away. Ryan is right, the points on S^1 with positive (resp negative) Y-coordinate are all identified. The space is like, {1, -1, N, S}. @Ryan, I believe it is the 1-skeleton. A loop inside a disk (disk in R^2) can be homotoped to a loop on the boundary S^1. Am I committing some obvious blunder here? $\endgroup$
    – Abhishek Parab
    Commented Nov 10, 2010 at 16:59
  • $\begingroup$ Well, the fundamental group of a 1-skeleton is free. The 2-skeleton provides the relations, as you have demonstrated. $\endgroup$
    – Ryan Reich
    Commented Nov 10, 2010 at 17:04

1 Answer 1

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In fact, there is the following theorem: Every finite CW complex is weakly homotopy equivalent to a finite topological space, and vice versa.

For simplicial complexes, this theorem is realized by mapping a complex to its face poset, and using the correspondence between finite posets and finite topological spaces. In the other direction, one maps a poset to its order complex.

In general it is not easy to compute homotopy groups of a finite topological space. I know that there are some techniques in Jonathan Barmak's Ph.D. thesis.

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    $\begingroup$ +1: This is such a striking result that I am almost upset that no one told me about it in the algebraic topology classes I took in my youth. $\endgroup$
    – Pete L. Clark
    Commented Nov 10, 2010 at 15:25
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    $\begingroup$ You went to Chicago, right? It must have been before Peter May started to love this theorem; he taught a summer course on it in 2003. $\endgroup$
    – Ryan Reich
    Commented Nov 10, 2010 at 15:54
  • $\begingroup$ This result is fantastic! (This being my first course in Alg Topology, I didn't know weak homotopy equivalence definition, but looked up.) So now may I ask, how one calculates the fund group of a CW complex 'efficiently'? Unfortunately, Barmak's thesis seems to be in French. $\endgroup$
    – Abhishek Parab
    Commented Nov 10, 2010 at 17:05
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    $\begingroup$ Jonathan Barmak's thesis is definitely not in French. :) The abstract and introduction are in Spanish, though. As for how to compute the fundamental group of a concretely given CW complex, I can only give two tips: (i) use van Kampen wherever possible; (ii) think of the 1-cells as generators and 2-cells as relations. $\endgroup$
    – Dan Petersen
    Commented Nov 10, 2010 at 17:16
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    $\begingroup$ May has some notes on his webpage: math.uchicago.edu/~may/MISCMaster.html $\endgroup$
    – Dan Petersen
    Commented Nov 11, 2010 at 16:04

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