High-Dimensional Generalizations of Interlacings of Lines

Thus, the theory of nonsingular plane configurations of lines is quite different from the theory of interlacings of lines. This closely corresponds to the picture one sees in the topology of manifolds: it is known that the topology of manifolds of successive dimensions has far fewer common features than the topology of manifolds whose dimensions differ by 4.

In the topology of high-dimensional manifolds one even has precise constructions which embed various parts of $n$-dimensional topology in $(n+4)$-dimensional topology. In surgery theory this construction is multiplication by a complex projective plane; in knot theory it is the two-fold covering of Bredon; and in the theory of singularities it is the addition to our function of the sum of the squares of two new variables.

It seems that something similar occurs in the theory of projective configurations. Interlacings of skew lines in three-dimensional space appear to be related to configurations of pairwise skew $(2k-1)$-dimensional subspaces in $(4k-1)$-dimensional space. One can define a linking number for oriented skew $(2k-1)$-dimensional subspaces of $(4k-1)$-dimensional space. Hence, all of the results on nonamphiheiral interlacings that were proved using linking numbers carry over to this multidimensional setting.

Moreover, there is a simple construction which to any such configuration associates a configuration of the same type with $k$ increased by 1. This construction preserves the linking numbers, isotopic configurations are taken to isotopic configurations, and perhaps to some extent one has an embedding of the theory of configurations of $(2k-1)$-dimensional subspaces of $(4k-1)$-dimensional space in the theory of configurations of $(2k+1)$-dimensional subspaces of $(4k+3)$-dimensional space. This gives rise to the possible development of a stable theory of projective configurations.

Here we shall give a description of this construction. As far as we know, it has not been published prior to the second version [2] of this paper, and it was the only original result of [2]. Our construction of a suspension is applicable not only to configurations of $(2k-1)$-dimensional subspaces of $(4k-1)$-dimensional space. It applies to any configuration of finitely many subspaces in projective space; it increases the subspace dimension by 2 and the ambient space dimension by 4.

We first recall the construction of the join of ordered configurations. Let $L_1$, ..., $L_r$ be subspaces of $\mathbb{R}P^p$, and let $M_1,\dotsc,M_r$ be subspaces of $\mathbb{R}P^q$. We imbed $\mathbb{R}P^p$ and $\mathbb{R}P^q$ in $\mathbb{R} P^{p+q+1}$ as skew subspaces. In the case of odd $p$ and $q$ the imbeddings should be chosen with care about orientations: the images, with their native orientations, should have positive linking numbers in $\mathbb{R} P^{p+q+1}$. Let $K_1,\dotsc,K_r$ denote the subspaces of $\mathbb{R} P^{p+q+1}$ such that $K_i$ is the union of all lines which intersect $L_i$ and $M_i$. We call the configuration of subspaces $K_1,\dotsc,K_r$ the join of our two configurations. ³

By the suspension of an arbitrary configuration $L_1,\dotsc,L_r$ of subspaces of $\mathbb{R}P^p$ we mean its join with a configuration of $r$ generatrices of a (one-sheeted) hyperboloid in $\mathbb{R}P^3$ with positive linking number (i.e., its join with the configuration of lines in $\mathbb{R}P^3$ corresponding to the interlacing which we denoted by $\langle +r\rangle )$. Since any two lines of the interlacing $\langle +r\rangle$ are isotropic, it follows that one can find an isotopy of this interlacing which permutes the lines in an arbitrary way. Hence, the join with an ordered configuration of subspaces $L_1,\dotsc,L_r$ in $\mathbb{R}P^p$ does not depend on the order. Thus, the suspension is well defined (up to rigid isotopy) for unordered configurations.

Two configurations of $k$-dimensional subspaces of $\mathbb{R}P^{2k+1}$ are said to be stably equivalent if there exists $N$ such that their $N$-fold suspensions are rigid isotopic. Mazurovskiui[14] has shown that this stable equivalence shares properties which are common for various stable equivalences mentioned above. Namely, Mazurovski [14] has proved that for $k>0$ any configuration of $\le k+5$ disjoint $(k+2)$-dimensional subspaces of $\mathbb{R}P^{2k+5}$ is rigidly isotopic to the suspension of a configuration of $k$-dimensional subspaces of $\mathbb{R}P^{2k+1}$, and, if there are $\le k+2$ subspaces in the configurations, then rigid isotopy of the suspensions is equivalent to rigid isotopy of the original configurations of $k$-dimensional subspaces of $\mathbb{R}P^{2k+1}$.

This stabilization theorem was used by Mazurovskiin [14] for obtaining the rigid isotopy classification of nonsingular configuration of six $(2k-1)$-dimensional subspaces in $\mathbb{R}P^{4k-1}$. He proved that when $k>1$ such configuration is defined up to rigid isotopy by the linking numbers. Recall that this is not the case for $k=1$, that is for configurations of lines in the 3-space. Suspension makes configuration $M$ shown in Figure 25 and its mirror image $M'$ rigidly isotopic. Recall that $M$ and $M'$ are distinguished by the Kauffman bracket polynomial. Thus, there is no generalization of the Kauffman bracket to high-dimensional nonsingular configurations which would be preserved under suspension.

Then Khashin and Mazurovski[15] proved that

Two nonsingular configurations of $k$-dimensional subspaces of $\mathbb{R}P^{2k+1}$ are stably equivalent if and only if they have the same linking numbers of the subspaces.

This means that there exists a bijection between the set of the $k$-subspaces of the first configuration and the set of the $k$-subspaces of the other configuration such that the linking numbers of the corresponding subspaces are equal.

Algebraic techniques developed for that was used in [15] also for obtaining the following two results about interlacings of skew lines in the 3-space:

Two isotopy join interlacings are rigidly isotopic if and only if they have the same linking numbers (and hence stably equivalent).

An interlacing of skew lines which has the same linking numbers as the configuration of disjoint generatrices of a (one-sheeted) hyperboloid in $\mathbb{R}P^3$ is rigidly isotopic to this configuration of generatrices.