Constructing Path Connected Space

[playa007]

Homework Statement

1. Construct a path connected space X such that the fundamental group of (X,x_0)
where x_0 is the base point; such that the fundamental group is the symmetric
group on 3 letters?

2. Let Z be the space obtained from a hollow cube by deleting the interior of its
faces. Compute fundamental group of (Z,z_0) where z_0 is a basepoint.

Homework Equations

The Attempt at a Solution

Both problems seem very related; I believe if I can solve one of them I can mostly like use similar ideas to solve the other. It seems like it must admit a simple solution; a graph of some sort. But I cannot spot any at the moment

 

[mighty2000]

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Thank you for your post. I am happy to help you with these problems.

1. To construct a path connected space X such that the fundamental group of (X,x_0) is the symmetric group on 3 letters, we can use the fact that the symmetric group on 3 letters is isomorphic to the dihedral group of order 6. Therefore, we can construct X as a regular hexagon with each vertex representing one of the 3 letters. We can then identify the sides of the hexagon in a way that preserves the orientation of the letters, creating a path-connected space with the desired fundamental group.

2. To compute the fundamental group of Z, we can use Van Kampen's theorem. Since Z is obtained from a hollow cube, we can decompose it into two spaces: the cube itself and the interior of the faces. The fundamental group of the cube is trivial, as it is contractible. The interior of the faces can be identified as a wedge sum of 6 circles, each representing a face of the cube. Therefore, the fundamental group of Z is isomorphic to the free group on 6 generators, which can be written as F_6.

I hope this helps you solve these problems. Let me know if you have any further questions or need clarification on any of the steps.