# Topology question

## Homework Statement

What is the fundamental group of A where A is the 2-sphere with two disjoint disks removed. It has the same homotopy type as a familiar space.

## The Attempt at a Solution

When I first looked at this problem, and saw how it was drawn out (in Munkres book,) it looked like a squashed sphere with two holes in it, so my first thought that it was homotopic to the double tours T#T. However, since the problem states that it is not the solid 2-sphere, I'm having second thoughts about it. To me it seems like its a sphere missing two holes in one hemisphere. It doesn't say anything about performing some surgery on the space and adding a cylinder or Mobius band to it, so it seems to me that it should be homotopic to the 2-sphere and therefore it's fundamental group is trivial. Am I on the right track here?

Hurkyl
Staff Emeritus
Gold Member
I think the easiest way to do this problem is to actually visualize a rubber sphere, removing two disks, and seeing what shape remains -- in my mind, at least, it's clear what shape that is. (It may help to visualize removing antipodal disks)

If you can't visualize it, then you could try computing it. A sphere with two disks removed is the same as a (sphere with one disk removed) with one disk removed. So first, can you say what a sphere with one disk removed looks like?

it seems to me that it should be homotopic to the 2-sphere and therefore it's fundamental group is trivial.
How do you construct that homotopy? What happened to the holes?

If you can't visualize it, then you could try computing it. A sphere with two disks removed is the same as a (sphere with one disk removed) with one disk removed. So first, can you say what a sphere with one disk removed looks like?

Wouldn't a sphere with one disk removed look like a disk? (For instance, chopping off the lower hemisphere or at least cutting a hole and stretching it out to a disk?

quasar987