nice space with wild fundamental group

Question

I would like to know an example of nice space with very strange fundamental group. With simplices and similar things I only get finitely presented groups.

Edit. I know from comments that Hawaiian earring has uncountable fundamental group. This shows that it is not a simplicial complex with one edge for every "loop". This contrasts the intuition given by its picture.

What is the intuitive reason for the Hawaiian earring not to be a simplicial complex?

Answer 1

There is a dichotomy for fundamental groups of locally path connected, compact metric spaces (called Peano continua).

Theorem (Shelah/Eda): If $X$ is a locally path connected, compact metric space, then either

1) $X$ is semilocally simply connected and $\pi_1(X,x_0)$ is finitely presented

or

2) $X$ is NOT semilocally simply connected and $\pi_1(X,x_0)$ is uncountable (and typically pretty wild).

It is pretty interesting that there is no room in the middle for fundamental groups with countably infinite generators. To address your specific question, every simplicial complex is semilocally simply connected. Since the Hawaiian earring does not have this property, it cannot be (or even have the same homotopy type as) a simplicial complex (finite or infinite)

Examples:

• Any finite simplicial complex satisfies (1)
• The Hawaiian earring (along with the Sierpinski carpet and Menger curve) satisfies (2) and has a wild fundamental group
• The infinite torus $\prod_{n=1}^{\infty}S^1$ satisfies (2) and has fundamental group $\prod_{n=1}^{\infty}\mathbb{Z}$. This group - called the Specker group- is uncountable but the algebraic structure is not quite as wild.

If you are interested in understanding the "wildness" of the fundamental group of the Hawaiian earring, I give an introduction to it in this blog post:

http://wildtopology.wordpress.com/2013/11/23/the-hawaiian-earring/

Another easy-to-construct space in $\mathbb{R}^3$ that has a "wild" fundamental group is the harmonic archipelago, which I describe in another blog post:

http://wildtopology.wordpress.com/2014/05/01/the-harmonic-archipelago/