Problem set 3 due Tuesday, Feb 28

Unlike previous problem sets, this one is not a collection of exercises from the material in the lectures, instead it gives you practice computing fundamental groups.

1. Show that if \(X\) is path connected, then for any two points \(x_0\) and \(x_1\) in \(X\), \(\pi(X,x_0)\) is isomorphic to \(\pi(X,x_1)\). More generally, show that in any groupoid \(G\), if there is a morhpism from \(x\) to \(y\), then the group of automorphisms of \(x\), \(\hom_G(x,x)\), is isomorphic to \(\hom_G(y,y)\).

   Thanks to this result in the following questions about fundamental groups you are free to pick any base point you’d like.

2. Compute the fundamental group of the space \(X\) consisting of a circle and one of its diameters by using van Kampen in a different way from what we did in class: take \(U\) and \(V\) to be the left and right halves of \(X\) (here, I’m picturing the diameter as horizontal), so that \(U \cap V\) has 3 pieces.
3. Compute the fundamental group of a sphere with a diameter.
4. Compute the fundamental group of the Klein bottle. One construction of a Klein bottle is to take a rectangle and indentify the opposite sides as shown in the figure below. Anopther construction is to start with a Moebius band, whose boundary is a single simple closed curve \(C\), and sew in a disk by identifying its boundary (which is just a circle) with \(C\). Choose either one of those constructions or another if you prefer, just say which one you’re using.

1. The compact orientable surface of genus \(g\) is like a torus but with \(g\) holes instead of just one (below there is a picture a surface of genus 3). Compute it’s fundamental group.