Higher Order π Groups, Suspension

Suspension

The suspension of a space X is X cross the unit interval, followed by a quotient map that shrinks X cross 0 to a point and X cross 1 to a point. It takes some imagination to see why this is called a suspension. At the middle of the unit interval, t = ½, the suspension looks like X. Move towards t = 0 or t = 1 and X shrinks down to a point. This may remind you of a hammock hanging from two points, that looks like X along every cross section. Thus X is "suspended" from two points.

If X is a pointed space, with base point x₀, apply another quotient map that shrinks x₀ cross I to a point, which becomes the base point of the suspension. Continuing our hammock analogy, x₀ defines a thread that runs from one end of the hammock to the other. Shrink this thread down to a point. The hammock now hangs from one point. If you are an ant, away from x₀, and away from either end, it looks like you're living in X cross I as usual.

Let's illustrate with the zero dimensional sphere, which is simply two points. Stretch these points into two threads and sew them together at either end. This is homeomorphic to a circle. If x₀ is designated as the base point, then one of the two threads shrinks to a point. The other thread closes up to become a circle. Therefore the suspension, or the pointed suspension, of S⁰ = S¹.

Let X = S¹, the traditional circle. Cross X with I to get a tube, then close the ends to find something homeomorphic to the sphere. The radius of X starts at 0 on the left, rises to 1 in the middle, and shrinks back down to 0 again at the right. In general, the suspension of Sⁿ = Sⁿ⁺¹.

A similar result holds for pointed suspensions. Shrink an arc on the surface of the sphere down to a point, stretching the nearby fabric along the way, and the result is homeomorphic to the original sphere. This is pretty intuitive, but let's take two steps closer to a proof.

Let the arc be the semi-equator, running through the western hemisphere. Pull the two meridians together, squeezing the western hemisphere down to a thin vertical slice, and stretching the eastern hemisphere commensurately. This shrink/stretch map is continuous, and homeomorphic to the original sphere.

Now all of North and South America are trapped between two lines of longitude that are only a few degrees apart. Our goal is to shrink a short arc, running along the equator, down to a point. This arc was once half the equator, but now it is only a few degrees across. Let this arc be the diameter of a small circle embedded in the sphere. Squash this circle into a figure 8. The diameter shrinks to a point, and lines just above and below the diameter shrink to very short arcs. The rest of the sphere, outside the circle, stretches to compensate. This shrinking and stretching reduces to 0 as you approach the top and bottom of the circle. The map is continuous, and the result is a sphere. Combine this with the previous map, and the pointed suspension of Sⁿ = Sⁿ⁺¹.

Reorientation

Often the suspension is pictured vertically, rather than horizontally. A space X is spread through a vertical distance corresponding to I, then the top and bottom shrink to two points, closing up the suspension. If we are dealing with pointed spaces, the base point of X, which has become a base line in X cross I, shrinks to a point, thus merging the top and bottom. The operation is the same; I've just turned the image 90 degrees. This orientation is quite common, even though it doesn't reflect the physical notion of hanging a hammock between two trees in the back yard. I just didn't want you to get confused.