J. F. Adams' Deformation Retract

The question of exactly what minimization problem a soap film solves is a tricky one. Reifenberg [1] considered soap films to be sets that spanned a boundary in a homological sense. Adams published an appendix to [1] in which he described this soap film as an example of a set we would like to call a soap film, but which does not span its boundary in any homological sense. This film, on an unknotted wire, has a deformation retract to the wire. A retract from a set Y (the film) to a subset X (the wire) is a continuous map from Y to X which is the identity map on X. A deformation retract of Y to X is a homotopy from the identity map on Y to a retract of Y to X, with the image of the homotopy remaining in Y. It is, in fact, a strong deformation retract since the homotopy can be chosen to always be the identity map on the wire. The existence of this deformation retract shows that the soap film cannot be said to span the wire in any homological sense (at least if one stays with a wire-embedded-in-Euclidean-space model of soap films).

The accompanying Java applet can help visualize the homotopy. It starts with one of the flat disks, the left, say, retracted horizontally along the midplane to the wire, stretching the film behind it into a double layer. Then the film retracts up and down between the wire and the triple lines, sweeping out the right disk, then back over to sweep out the left disks.

Another way to visualize the deformation retract is to inverse image of each wire point in the final retract. The Java applet shows these in green. Each inverse image is a curve emanating from the point and snaking around the surface until it ends in a loop. The inverse image of the wire points on the midplane also include the flat disks on their respective sides. The homotopy may be visualized as the end loops shrinking down and back along their curves to their base points on the wire.

Applet Instructions:

Mouse: The surface may be rotated, translated, scaled, or spun by dragging your mouse over it. The buttons at the top left control which mode is active.
Fineness: The buttons at the top right control the fineness of the decompostion of the surface for drawing purposes. You can adjust the fineness to suit the speed of your computer.
Inverse image: The slider at the bottom controls display of the inverse image of a boundary point. The inverse image is displayed in green.
Deformation: The slider at the left controls the degree of retract. For visibility of the initial stages of the retract, a line segment is shown sweeping between the left rings, pushing the disk in front of it until it hits the wire, and leaving a double layer behind it.
Height: The slider at the right controls the separation of the rings.
[1] E. R. Reifenberg, "Solution of the Plateau Problem for m-dimensional surfaces of varying topological type," Acta Math. 104 (1960) 1-92.
Author: Kenneth A. Brakke, Mathematics Department, Susquehanna University. Last modified: May 1, 2001.