Simply-Connected Spaces

We say that a space is “simply-connected” if any closed curve with is homotopic to a constant curve that stays at the single point . Intuitively, this means that any loop in the space can be “pulled tight” without getting caught up on any “holes”.

It turns out that this is equivalent to saying that every closed curve is the boundary of some parameterized square. Indeed, consider the following diagram I’ve drawn with the help of Geogebra:

This is a picture of the homotopy cylinder. The domain of a curve is the interval , so the domain of the homotopy cylinder is the square . I’ve labeled the sides to describe what the homotopy does to them: the lower edge follows the curve ; the upper edge is the constant point ; the two sides are also constant at , meaning that we’re holding the curve’s ends fixed as we perform the homotopy. And so the homotopy is exactly a continuous (or smooth) map from the square into our space, and the boundary of the parameterized square is exactly the curve . The converse — that any parameterized square can be homotoped to look like this — shouldn’t be hard to see.

So what does this mean for homology? Well, for cubic singular homology it means that is exact if is simply-connected. Indeed, if is a closed -chain, then it must be made up of a formal sum of curves. Any curve which isn’t already closed must have a start and an end, and the end must be the start of another curve, or else the boundary points of wouldn’t cancel off. We can break up — possibly non-uniquely — into a collection of closed curves, each of which is the boundary of some parameterized square, by the above argument. Thus is itself the boundary of this collection of squares; since all closed -chains are exact, the first homology vanishes.