# 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.