The task is: Compute the fundamental group of the topological space obtained by identifying the four vertices of a square.

So we identify the vertices with the same letter. Can we say something about the orientation of sides ?

How can we use Van-Kampen theorem here ?

  • $\begingroup$ This questions is very unclear. Are you literally asking for the space where all vertices are identified but nothing is done to the edges? $\endgroup$ – Cheerful Parsnip Nov 20 '13 at 14:03
  • $\begingroup$ Yes, this is exactly the exercise. $\endgroup$ – derivative Nov 20 '13 at 14:04
  • $\begingroup$ Please ensure that your question is fully understandable without the title. The title's purpose is to attract people to your question (and to make it easily searchable). Once it has done that, it's job is done. The question should be self-contained. $\endgroup$ – Lord_Farin Nov 20 '13 at 15:48

I wouldn't use Van Kampen. I would just note that this space deformation retracts to the union of the two diagonals of the square which is an X shape with the four ends glued together. This in turn is homotopy equivalent to a wedge of three circles by contracting one of the edges. Do you know the fundamental group of this?

Or to use Van Kampen, a neighborhood of the perimeter has free fundamental group generated by the four edges. Gluing in the interior of the square adds the relation that the product of these four generators is trivial. So we have $\langle a,b,c,d\,|\, abcd=1\rangle$, which isomorphic to a free group on three generators.

  • $\begingroup$ The deformation retraction is by far the simplest approach. $\endgroup$ – Dan Rust Nov 20 '13 at 14:11
  • $\begingroup$ you can express one letter with the other three, that's why 3 generators, right ? $\endgroup$ – derivative Nov 20 '13 at 14:19
  • $\begingroup$ @derivative: yes. $\endgroup$ – Cheerful Parsnip Nov 20 '13 at 14:36

I like to see this as part of a general question: Let $X$ be a topological space with a subspace $A$ which is discrete and $i:A \to X$ is a cofibration. Let $f:A \to B$ be a function to another set, and let $Y$ be the space obtained from $X$ by identifying the points of $A$ according to $f$. (For example, $B$ could be a singleton.) We wish to describe the morphism $\phi: \pi_1(X,A) \to \pi_1(Y,B)$ of groupoids. It is a consequence of the groupoid version of the Seifert-van Kampen Theorem that $\phi$ is a universal morphism in the sense of Higgins' book Categories and Groupoids. If $B$ is a singleton, then we get the "universal group" of the groupoid. The universal group of a groupoid gives the left adjoint to the inclusion of the category of groups into the category of groupoids.

In particular, if $X$ is contractible, then the universal group of $\pi_1(X,A)$ is a free group.


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