Transversality in Configuration Spaces and the “Square-Peg” theorem
We prove a transversality “lifting property” for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclidean space by an arbitrarily -small variation of the initial submanifold, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide attractive proofs of the existence of a number of “special inscribed configurations” inside families of spheres embedded in using differential topology. For instance, there is a -dense family of smooth embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a -dense family of smooth embedded -spheres in where each sphere has a family of inscribed regular -simplices with the homology of .
I Introduction††margin: I sect:intro
Given a simple closed curve (a Jordan curve) in , can we find four points on that form a square? This question was posed by Toeplitz in 1911 Toeplitz:1911ta () and it has drawn the attention of many mathematicians over the intervening century. Thinking of the Jordan curve as a “round hole”, the problem has been affectionately dubbed the “square-peg problem.” We say that the square is inscribed in when the vertices lie on the curve. We do not require that the square lie entirely in the interior of the curve. Progress on the square-peg problem has chiefly been extension of the class of simple closed curves for which the square can be found. (The interested reader can find a number of survey articles such as MR1133201 (); Mat-SP-survey (); Pak-Discrete-Poly-Geom (). )
Our goals are different. First, by placing the problem in the context of configuration spaces and their subspaces, we have opened up a set of tools from differential topology that allow fresh viewpoints through some powerful methods. Our conclusions include the previous work and show how differentiability assumptions can deliver a strong sense in which squares appear generically. The use of the multijet transversality theorem Golubitsky:1974iu () is new and holds promise for the application of differential topological methods to other configuration problems.
Here is the heart of our method: If we consider the (compactified) configuration space of 4-tuples of points in the plane as an 8-dimensional manifold with boundary (and corners), then Toeplitz’s question can be rephrased more simply as a question about the intersections of the 4-dimensional submanifold of 4-tuples of points on , called , with the 4-dimensional submanifold of squares in .
We can see with a little effort that for a standard ellipse, these submanifolds intersect in four points corresponding to cyclic relabelings of a single inscribed square. It is therefore natural to try to show that squares are transverse to inscribed configurations in the ellipse and use an isotopy from the ellipse to to connect the square on the ellipse to a cobordant family of squares on the target curve.
This program requires us to face a few technical obstacles. First, the square might shrink away during the isotopy. We overcome this obstacle by analyzing the (compactified) boundary of our submanifolds of inscribed configurations and showing that, in a precise sense, curves do not admit infinitesimal squares. Second, we do not know that the submanifold of squares is transverse to the submanifold of inscribed configurations on . We may vary the submanifold of inscribed configurations using the standard transversality theorem for manifolds to make it transverse, of course, but there is no a priori guarantee that the varied submanifold consists of inscribed configurations on any single curve. We deal with this problem by an application of the multijet transversality theorem Golubitsky:1974iu (). Third, it turns out to be the case that the four intersections of the submanifold of squares with the submanifold of inscribed quadruples on the ellipse alternate sign. To count squares we must mod out by cyclic relabeling of vertices and pass to intersection theory.
The method we use for squares is an example of a general approach to such “special inscribed configuration” problems: Show that the configurations one is looking for form a submanifold of configuration space to establish smoothness, prevent “shrink outs” by showing that is boundary-disjoint from the submanifold of inscribed configurations , find the (transverse) intersection of and explicitly in a base case, use our transversality theorem to conclude that a submanifold near the target submanifold also has . Finally, use standard methods to build a isotopy from to that is transverse to at every step of the way.
In addition to counting squares (Theorem 24), we show as another example of these methods that there is a dimensional family of inscribed simplices of any edgelength ratio in a generic embedding of in (Theorem 28).
It is important to note that while our results provide a unified and attractive view of this family of theorems about special inscribed configurations, they do not directly address the remaining open territory in Toeplitz’s question: We give, in the Appendix, an extension of our results to prove that there exists at least one square on any embedded curve of finite total curvature without cusps, but this class of curves is certainly less general than the family of curves for which Stromquist proved the square peg theorem in MR1045781 ().
Ii Configuration Spaces††margin: II sect:config
The compactified configuration space of points in is the natural setting for both the square-peg and inscribed polygon problem. A reader familiar with configuration spaces may skip much of this section. However we recommend paying attention to the notation we have used for the spaces, points in the spaces and the strata. Definition 2, Definition 3, and Remark 5 are particularly useful. This section provides a brief overview of (compactified) configuration spaces. There are many versions of this classical material (see for instance MR1259368 (); MR1258919 ()). We follow Sinha newkey119 () as this gives a geometric viewpoint appropriate to our setting.
Definition 1.††margin: 1 def:openconfig
Given an -dimensional smooth manifold , let denote copies of , and define to be the subspace of points such that if . Let denote the inclusion map of in .
The space is an open submanifold of . Our goal is to compactify to a closed manifold with boundary and corners, which we will denote , without changing its homotopy type. The resulting manifold will be homeomorphic to with an open neighborhood of the fat diagonal removed. Recall that the fat diagonal is the subset of of -tuples for which (at least) two entries are equal, that is, where some collection of points comes together at a single point. The construction of preserves information about the directions and relative rates of approach of each group of collapsing points.
Definition 2 (newkey119 () Definition 1.3).††margin: 2 def:pijsijk
Let denote the number of ordered subsets of distinct elements of a set of size . Given an ordered pair of , let be the map that sends to , the unit vector in the direction of . Let be the one-point compactification of . Given an ordered triple of distinct elements in , let be the map which sends to .
To define configuration spaces for points in an arbitrary smooth () manifold , we embed in so that is a subspace of . We then compactify the space as follows:
Definition 3 (newkey119 () Definition 1.3).††margin: 3 def:config
Let be the product . Define to be the closure of the image of under the map
If is smoothly embedded in , then is smoothly embedded in and we define to be the closure of in . In this case, we will refer to as for convenience; we denote the boundary of by .
We now summarize some of the important features of this construction, including the fact that does not depend on the choice of embedding of in .
Theorem 4.††margin: 4 thm:config
is a manifold with boundary and corners with interior having the same homotopy type as . The topological type of is independent of the embedding of in , and is compact if is.
The inclusion of in extends to a surjective map fron to which is a homeomorphism over points in .
Remark 5.††margin: 5 rmark:notation
When discussing points in or , it is easy to become confused. We pause to clarify notation.
A point in is denoted by , where each .
Points in are also denoted by , where and each . (It will be clear from context which is meant.)
A point in or , is denoted .
At times, we will need to distinguish between the various entries of or . In general,
where , and gives the corresponding set of values in and .
The space may be viewed as a polytope with a combinatorial structure based on the different ways groups of points in can come together. This structure defines a stratification of into a collection of closed faces of various dimensions whose intersections are members of the collection. We will need to understand a bit of the structure of this collection, which is referred to as a stratification of .
Definition 6 (newkey118 () Definition 2.4).
A parenthesization of a set is an unordered collection of subsets of such that each subset contains at least 2 elements and two subsets are either disjoint or one is contained in the other. A parenthesization is denoted by a nested listing of the using parentheses. Let denote the set of parenthesizations of , and define an ordering on it by if .
For example, for , represents a parenthesization whose subsets are and while represents a parenthesization whose subsets are and .
We identify each parenthesization of with a closed subset of in our stratification of . The idea is that all the points in each collapse together, but if , then the points in collapse “faster” than the points in . Formally, this becomes the following condition: Let be a point in . Then if
if and only if for some .
(and hence ) if and only if , and .
Sinha proves that a stratum described by nested subsets has codimension in . In the previous example has codimension 1, while and have codimension 2.
We notice that the definition of the does not depend on the . In fact, for connected manifolds of dimension at least , the combinatorial structure of the strata of depends only on the number of points. Regardless of dimension, this construction and division of into strata is functorial in the sense that
Theorem 7 (newkey119 ()).††margin: 7 thm:functor
An embedding induces an embedding of manifolds with corners called the evaluation map that respects the stratifications.
Corollary 8.††margin: 8 cor:smooth
Let be a smooth diffeomorphism. Then the induced map of configuration spaces is also a smooth diffeomorphism (on each face of ).
This is an immediate corollary of the previous theorem. ∎
Any pair , of disjoint points in has a direction associated to it, while every triple of disjoint points , , has a corresponding distance ratio . One way to think of the purpose of is that it extends the definition of these directions and ratios to the boundary.
Theorem 9 (newkey119 () or newkey118 () Theorem 2.3).††margin: 9 thm:configgeom
Given , in any configuration of points each pair of points , has associated to it a well-defined unit vector in giving the direction from to . If the pair of points project to the same point of , this vector lies in .
Similarly, each triple of points , , has associated to it a well-defined scalar in corresponding to the ratio of the distances and . If any pair of projects to the same point in (or all three do), this ratio is a limiting ratio of distances.
The functions and are continuous on all of and smooth on each face of .
Iii Special Submanifolds of Configuration Spaces
We are interested in three special submanifolds of particular configurations defined by geometric constraints. First, we consider the configuration space of points on a curve.
Let be a -smooth embedding of in , with the evaluation map on configuration spaces. We abuse notation by using to mean either the embedding or its image in . Similarly, we use to mean either the evaluation map or its image — the compactified configuration space of points on the simple closed curve .
By Theorem 7 we know that is a submanifold of and with the stratifications respected. The coordinates for are similar to those described in Theorem 9, as they are the image of the coordinates under . Volic MR2300426 () and Budney et al. newkey118 () have detailed descriptions of the coordinates for codimension 1 strata. To give an example, observe that the map takes to . If we consider the stratum where say , and degenerate to a point in , then is a configuration of points plus the and information for , and . In we get a configuration of points on plus the directions of approach of the colliding and the relative distances , , and so forth. The are unit tangent vectors to . If and approach equally from opposite sides, then in the limit , so the obey the relations
In the values of are in and are mapped to by . Thus, while the exact values of the unit tangent vectors and are unknown for two colliding points on , they must differ by .
In the case of the circle, the cyclic ordering of points along determines connected components of . Note that some strata are empty in the boundary of each connected component of . For instance, in the component of where points , , and occur in order along , if and come together, either or must collapse to the same point. Thus the stratum is empty on the boundary of this component. We will focus on one of these connected components:
Definition 11.††margin: 11 def:ccng
Let denote the component of where the order of the points matches the cyclic order of these points along according to the given parametrization of .
We now consider another submanifold – this one with a more interesting structure.
Let the subset of square-like quadrilaterals for be the subspace of squares in , and for , the subset of where and . That is, is the space of quadrilaterals in with equal sides and equal diagonals. ††margin: 12 def:slq
Proposition 13.††margin: 13 prop:slq is submanifold of r2
The space is an orientable submanifold of , and the (point-set) boundary of satisfies .
Let be a point in , and consider the mapping given by
This mapping is smooth and is the preimage of the point . We show that
is onto at points by showing has four linearly independent rows. We denote a tangent vector at by , where each is a tangent vector at . (Here we suppress the information on the strata.)
Let denote a vector at as in Figure 2, define and consider
To compute the limit, let us consider a typical quotient term involved:
Next divide by . We can ignore terms in the numerator with because they will vanish in the limit. We rearrange to get:
Taking the limit as , we get:
where , and is the angle between vector and the vector given by .
Similar computations give an explicit form to ; suppose . Then
Since the angles made by and the sides and diagonals of a given quadrilateral cannot be chosen so that all cosines involved vanish at once, does not vanish on .
Analogous variations , and at , , and respectively, lead to the following expressions:
After some elementary row operations, one finds that carefully chosen variations at , , , and will give four linearly independent vectors at points in . It follows from the Preimage Theorem of Guillemin:2010ti () that and the interior of is a submanifold of .
The boundary points in are where the points of a configuration come together, along with the directions of collision and ratios of the sides. There is no difficulty in the plane, where the ratios in the definition of may be smoothly extended to the boundary. The boundary is contained in the boundary face of , and, in fact, the map is transverse to on this boundary. (For the sake of brevity we have omitted the details.) Thus in this special case, is actually a submanifold with boundary of , the larger manifold with boundary.
The (pointset) boundary of in contains both “infinitesimal” squares and configurations in the face of , where the diagonals are equal to zero while the sidelengths remain equal and nonzero. Such collisions lead to square-like quadrilaterals that are four-fold covers of an interval. We may certainly extend the map to this face, but here we run into trouble: Since any configuration on the has equal sidelengths and equal diagonals, the map is not transverse to when restricted to this boundary face, and our argument does not show that is a submanifold with boundary of , the larger manifold with boundary. ∎
We next state a useful corollary of these detailed computations. Recall (Guillemin:2010ti ()) that if is transverse to and and are oriented, the orientation on at is constructed by appending a positively oriented basis for the “horizontal” subspace of to a basis for the “vertical” subspace . The vertical basis is considered positively oriented if the combined basis is a positively oriented basis for . We will be interested later in the free and properly discontinuous action of on and on that cyclically permutes , , and . Let be the map corresponding to the generator of for this action. It is clear from the definition of that descends to a map from to .
The map reverses orientation on both and if is odd, and preserves orientation on both and if is even. ††margin: 14 prop:orientation
We first note that , and recall that a tangent vector at is denoted by , where is a tangent vector at .
To prove the proposition, we now construct some specific variations of quadrilaterals in that will behave nicely under the action. For squares in the plane, Figure 3 shows the construction of two types of tangent vectors to at . The first three tangent vectors are of the form , and . Note is shown at the left in the figure and is perpendicular to . Assume that has and . As shown in the figure, we can arrange to have
The fourth tangent vector is shown at the right in Figure 3 and has
Working out the directional derivatives of , , , and in these directions, we see that restricted to the span of , , , and looks like the matrix:
where the entries represent nonzero values that we don’t need to compute.
Now we make a similar construction for nonplanar configurations in . Assume the square-like quadrilateral has sides of length etc., and diagonals have length . Consider the situation shown in Figure 4. Let us focus on edge for convenience. At the plane determined by , , and has normal vector, say . Consider the tangent vector . Since is perpendicular to vectors and , the directional derivatives of the lengths of these edges in this direction are zero. On the other hand, since the tetrahedron is not a planar square, is not in the plane normal to , so the directional derivative of is nonzero. We can now find some scalar multiple of so that . This implies that .
We can make a similar argument at vertex . Let be a normal vector of the plane and find parallel to so that and . A similar argument at yields a vector with while preserving all other edgelengths to first order. Scaling appropriately, we can arrange to have .
As shown in Figure 4 at right, we can also find a tangent direction so that while the directional derivatives of all other edgelengths vanish. This choice gives . Taken together, we have constructed a subspace of given by on which
Using these bases, we can now compute the effect of the action on the orientation of and . First, observe that the tangent space to contains of four copies of and that reordering these from to requires swaps of basis elements. Thus is orientation preserving or reversing on as is even or odd.
Now take any positively oriented basis for and extend it by a basis so that maps onto the tangent space to in such a way that the image of is positively oriented with respect to the orientation of . We want to know whether is positively oriented. We know that the combined basis is positively or negatively oriented in as is even or odd. It remains to show that maps onto the tangent space for so that the image is positively oriented. This comes down to an explicit calculation of determinants.
For a planar configuration , we use the basis , , , constructed above. We can compute that, on the space , we have:
where again, represents a value we don’t need to compute. This is a matrix of positive determinant, as desired. For a non-planar configuration in , we use the basis ,