CHAPTER 3

Suppose that you are standing on a geometrical object. You toss a lasso and try to shrink it to a single point at your feet (the lasso must stay on the ground). If there is no pool of water on this grassland the lasso will smoothly converge at your feet. Imagine however, that there is a pool of water which your lasso is enclosing (We assume that the interior of the pool does not belong to our geometrical object). In this case you cannot bring the lasso to a single point without getting it wet. The fundamental group or the first homotopy group of the geometrical object measures the degree of the possibility in shrinking the lasso to a point.
On the other hand the second homotopy group of a geometrical object measures the degree to which one can shrink a large piece of cloth, spread out with its border being gathered at a single point, to that point while keeping the cloth always in the object.
If the above explanation gives you some idea of what is going on, you might skip this chapter and proceed to the next one.