An Interior Fixed Point Property of the Disc

Abstract

1. STIRRING THE COFFEE. At the end of this evening's meal, stir your cup of coffee and watch the motion of the top of the coffee. You will probably notice that there seem to be points where the coffee is not moving. The Brouwer Fixed Point Theorem, implies that such points must always exist. To be precise, suppose you could set the whole top surface of the coffee in motion at once. Then after a moment, every point of the surface would have moved. We could think of this motion as defining a map, that is a continuous function, f: D -> D of the closed disc (the top surface) to itself. But Brouwer proved that the disc has the fixed point property, that is, that every map from D to itself has a fixed point: a point p E D such that f(p) = p. This contradiction shows that there is a point on the surface where the coffee is not moving. You will probably observe some fixed points, but they may turn out to be difficult to detect. Since Brouwer's Theorem says nothing about the location of the fixed points, the fixed points might lie only on the boundary of the disc. If the way you stir your coffee produces a map with fixed points of this kind, it may be hard to convince you that Brouwer's Theorem is true because the entire (interior) surface of the coffee would be in motion, with fixed points only where the coffee touches the cup. If a map f: D --> D moves every point on the boundary of D, which we shall call S, then f must have a fixed point in int(D), the interior of D, since f must have a fixed point somewhere on D. But when f does have a fixed point on S, then f may or may not have additional fixed points in int(D). It is natural to ask what one can conclude about the interior fixed points just from knowing the restriction of f to the boundary S, which we denote by f IS. We will be concerned with the case where f IS takes S to itself. If f IS: S -> S has at least one fixed point, then there is a map G: D --> D with no interior fixed points such that GIS = f IS, as we will show in the next section. What is surprising is that, even if f IS: S --> S is smooth, that is, continuously differentiable, there may not be any smooth map G: D -> D with GIS = f IS that has no interior fixed points. In joint research with Helga Schirmer [1], we investigated when smooth extensions of maps on the boundary of a compact differentiable manifold must have interior fixed points. Our goal in this article is to explain these ideas in a concrete and easily-visualized setting, the 2-dimensional disc. Most of the things that can be done continuously in topology can be done smoothly also, because of general results about smooth approximation of continuous functions. But there are some exceptions, usually surprising, to that general principle. We will tell you about a new surprising exception.