1pt

# A differential approach to Maxwell-Cremona liftings

Oleg Karpenkov, Fatemeh Mohammadi, Christian Müller, and Bernd Schulze

###### Abstract.

In 1864, J. C. Maxwell introduced a link between self-stressed frameworks in
the plane and piecewise linear liftings to 3-space. This connection has found
numerous applications in areas such as discrete geometry, control theory and
structural engineering. While there are some generalisations of this theory to
liftings of $d$-complexes in $d$-space, extensions for liftings of frameworks
in $d$-space for $d\geq 3$ have been missing. In this paper, we introduce and
study differential liftings on general graphs using differential forms
associated with the elements of the homotopy groups of the complements to the
frameworks. Such liftings play the role of integrands for the classical notion
of liftings for planar frameworks. We show that these differential liftings
have a natural extension to self-stressed frameworks in higher dimensions. As
a result we generalise the notion of classical liftings to both graphs and
multidimensional $k$-complexes in $d$-space ($k=2,\ldots,d$). Finally we
discuss a natural representation of generalised liftings as real-valued
functions on Grassmannians.

###### Contents

  1. 1 Preliminaries
  2. 2 Differential liftings in $\mathbb{R}^{2}$
  3. 3 Differential liftings in $\mathbb{R}^{n}$
  4. 4 Differential liftings of polytopal embeddings of complexes
    1. 4.1 The $m$-frameworks
    2. 4.2 Polytopal embeddings of complexes
    3. 4.3 Differential liftings of $m$-frameworks
    4. 4.4 Stresses and force-loads, and relation to the case $m=1$
  5. 5 On liftings in general settings
    1. 5.1 Liftings on regular spheres
      1. 5.1.1 Definition of liftings
      2. 5.1.2 Co-dimension 0 obstacles cycle integration
      3. 5.1.3 Co-dimension 1 obstacles cycle integration
      4. 5.1.4 Co-dimension 2 obstacles cycle integration
      5. 5.1.5 Continuity of liftings
    2. 5.2 Grassmannian liftings

## Introduction

In 1864, J. C. Maxwell and later also L. Cremona showed that a planar
framework in $\mathbb{R}^{2}$ with a self-stress has a dual reciprocal diagram
and can be lifted vertically to a polyhedral surface in $\mathbb{R}^{3}$. The
present paper is dedicated to answer the following question: _What are the
analogues of liftings for self-stressed frameworks in $\mathbb{R}^{n}$?_ For
$n\geq 3$, the answer to this question was not known. In order to approach
this question, we construct a new natural extension of liftings for general
frameworks in $\mathbb{R}^{n}$ and for Rybnikov’s polyhedral self-stressed
complexes. This will provide the answer to the above problem.

The story begins with two papers by J. C. Maxwell [21, 22] and one by L.
Cremona [11] in which the authors founded the field of Graphic Statics. They
discovered that self-stressability of planar frameworks can be visualised by
piecewise linear functions whose singular set is the framework itself. This
theory is closely related to the duality between Voronoi diagrams and Delaunay
tessellations. The result of Maxwell and Cremona has important connections and
applications in rigidity theory, polyhedral combinatorics and discrete and
computational geometry; see e.g. [40, 2, 38, 9, 10, 15, 36, 37, 30]. In
particular, the technique of liftings has contributed to the solution of the
Carpenter’s Rule problem [6]. Liftings are also used in a well known proof of
Steinitz’s Theorem from polyhedral combinatorics [31]. (See also [12] and
[28].) Recently, F. Mohammadi and X. Wu in [25] introduced a link between
self-stresses and Chow rings on toric varieties related to convex polytopes.
There has also been a resurgence of interest in techniques from Graphic
Statics in architecture and structural engineering [24]. (In engineering, the
3-dimensional polyhedra projecting down to self-stressed frameworks in the
plane are commonly known as _discrete Airy stress function polyhedra_.)
Moreover, liftings have found modern applications in control theory [26, 27].
For general references about liftings in discrete and computational geometry,
we refer the reader to [39, Chapter 61], as well as to [41] and [35].

Let us mention what is currently known about generalisations of the Maxwell-
Cremona correspondence. In 1993, H. Crapo and W. Whiteley investigated an
extension of this correspondence to $3$-space in an unpublished manuscript
[8]. 3D Graphic Statics has recently been further investigated in the
engineering community (see e.g. [20]). In 1999, K. Rybnikov introduced self-
stresses on polyhedral $n$-complexes in $\mathbb{R}^{n}$ and studied liftings
of these objects [32, 33]. More recently, A. Ribó Mor discussed liftings for
planar self-touching frameworks [29]. In 2014, C. Borcea and I. Streinu proved
an analogue of the theorem of Maxwell and Cremona for planar periodic
frameworks [4, 5]. Finally, J. Erickson and P. Lin studied reciprocal diagrams
of toroidal frameworks in [13], and Z. Cooperband, R. Ghrist and J. Hansen
very recently gave a description of Graphic Statics for planar and non-planar
frameworks using cosheaf homology in [7].

Extensions of liftings for frameworks in $\mathbb{R}^{n}$ for $n\geq 3$ have
not yet been obtained. In this paper we give a construction for this case and
for more general cases. As was already done by K. Rybnikov in the literature
[32, 33], the liftings of $n$-complexes in $\mathbb{R}^{n}$ are defined on the
complement to $n$-complexes. Our idea is to consider these complements as the
set of path-connected components denoted as $\pi_{0}$. This consideration
gives rise to a natural generalisation of liftings to the cases of complexes
of arbitrary dimensions (in particular for frameworks in $\mathbb{R}^{n}$) and
to the case of non-planar frameworks. Here the elements of the higher
fundamental groups $\pi_{m}$ of the complement to the frameworks/complexes
will serve as the domain sets in the definition of ‘differential liftings’.

In this paper, we consider piecewise constant $m$-forms on the elements of
$\pi_{m}$ of the complement to the frameworks/complexes in $\mathbb{R}^{n}$
which we call differential liftings (see Definitions 2.3, 3.6, and 4.10).
Differential liftings satisfy a certain boundary condition at every vertex
induced by equilibrium conditions on the stress coefficients on edges incident
with this vertex. In the classical case the liftings of planar frameworks in
$\mathbb{R}^{2}$ are the integral graphs of the corresponding differential
1-forms. We show that for every framework in $\mathbb{R}^{n}$ with a self-
stress there exists a unique differential lifting (Theorems 2.4 and 3.7). This
provides the answer to the original question regarding generalisations of
liftings for self-stressed frameworks in $\mathbb{R}^{n}$. Further we extend
this result to the case of general polyhedral complexes (Theorem 4.11). We
briefly discuss generalisations of liftings in the last section of this paper.
It is interesting to note that such liftings can be represented as functions
on affine Grassmannians (see Subsection 5.2).

We work mostly with differential liftings rather than with usual liftings for
the following reasons. On the one hand the differential liftings are simpler
than usual liftings in the sense that they are piecewise constant over
$\pi_{0}$ rather than piecewise linear. So they are invariants of the elements
of $\pi_{0}$. Therefore, we prefer to use them when extending differential
liftings to $\pi_{m}$ as invariants of its elements. On the other hand, the
differential liftings provide the same information on self-stressability as
the classical liftings for planar frameworks. Indeed, the liftings are
obtained from differential liftings by integration of the latter.

This paper is organised as follows. We begin by providing the necessary
classical notions from Geometric Rigidity Theory and Graphic Statics in
Section 1. In Section 2, we approach the notion of liftings of self-stressed
frameworks from a new perspective using differential forms and introduce the
natural notion of differential liftings. In Section 3, we define this notion
for frameworks in $n$-dimensional space. Finally, in Section 4 we extend the
concept of differential liftings to the case of polyhedral complexes
introduced by K. Rybnikov in 1999. We conclude the paper in Section 5 with
discussions on how to extend standard liftings from differential liftings in
general settings.

##  1\. Preliminaries

Our paper is based on two fundamental notions which are self-stressed
frameworks (Definition 1.1) and liftings (Definition 1.2). Self-stressed
frameworks are realisations of graphs in $\mathbb{R}^{n}$ which are in static
equilibrium. Liftings are, broadly speaking, polyhedra in a higher dimensional
space whose vertical projections are given self-stressed frameworks.

######  Definition 1.1.

Let $G=(V,E)$ be an arbitrary finite graph without loops and without multiple
edges where $V=\\{v_{1},\ldots,v_{d}\\}$ denotes the set of _vertices_ and $E$
the set of _edges_. We denote by $v_{i}v_{j}$ the edge joining $v_{i}$ and
$v_{j}$.

  * •

A _framework_ $G(p)$ in $\mathbb{R}^{n}$ consists of a graph $G=(V,E)$ and an
injective map $p:V\to\mathbb{R}^{n}$. We often denote $p(v_{i})$ by $p_{i}$.
We say that there is an _edge_ between $p_{i}$ and $p_{j}$ if $v_{i}v_{j}$ is
an edge of $G$ and denote it by $p_{i}p_{j}$. Note that we require that all
points are pairwise distinct.

  * •

A _stress_ $\omega$ on a framework is an assignment of real scalars
$\omega_{ij}$ to its edges $v_{i}v_{j}$ which are called _stress coefficients_
(also known as _tensions_ and _compressions_ depending on the sign) with the
property $\omega_{ij}=\omega_{ji}$. We also set $\omega_{ij}=0$ if there is no
edge between the corresponding vertices.

  * •

A stress $\omega$ is called a _self-stress_ if the equilibrium condition

$\sum\limits_{\\{j\mid j\neq i\\}}\omega_{ij}(p_{i}-p_{j})=0$

is fulfilled at every vertex $i$. By $p_{i}-p_{j}$ we denote the vector from
the point $p_{j}$ to the point $p_{i}$.

  * •

A pair $(G(p),w)$ is called a _tensegrity_ if $\omega$ is a self-stress for
the framework $G(p)$.

  * •

A tensegrity $(G(p),w)$ (or stress $\omega$) is said to be _non-zero_ if there
exists an edge $v_{i}v_{j}$ of the framework that has a non-vanishing stress
coefficient $\omega_{ij}\neq 0$.

We recall in the following definition the classical notion of a polyhedral
lifting over a planar graph that has been put into relation with self-stressed
frameworks by Maxwell [22].

\begin{overpic}[width=130.08731pt]{fig/lifting.jpg} \end{overpic} Figure 1.
Polyhedral lifting of a planar framework (see Definition 1.2).

######  Definition 1.2.

Let $G$ be a planar graph and $G(p)$ a corresponding framework embedded as
planar straight-line graph in $\mathbb{R}^{2}$. Denote by $F$ the set of
connected components of $\mathbb{R}^{2}\setminus G(P)$; we call the elements
of $F$ the faces. The only unbounded face is called the infinite face.
A continuous and piecewise affine function $L:\mathbb{R}^{2}\to\mathbb{R}$ is
called _$($ polyhedral$)$ lifting_ of $G(p)$ if $L$ restricted to any face of
$F$ is an affine map. See Figure 1 for an illustration of the graph of such a
function $L$.

Note that all faces are open sets with a polygonal boundary.

The graph $(u,v,L(u,v))$ of a polyhedral lifting is therefore a polyhedral
surface whose edges and vertices vertically project to the given graph.

######  Theorem 1.3 ([22]).

Let $G$ be a planar graph and $G(p)$ a framework embedded in $\mathbb{R}^{2}$
without crossing edges. Then $G(p)$ has a non-zero self-stress if and only if
$G(p)$ has a non-trivial polyhedral lifting $L$.

######  Remark 1.4.

Let us mention the classical notion of the Maxwell reciprocal diagram. The
_Maxwell reciprocal diagram_ of a framework $G(p)$ in $\mathbb{R}^{2}$ is a
framework $G^{*}(p^{*})$ defined on the combinatorial dual graph $G^{*}$ such
that corresponding edges of the frameworks $G$ and $G^{*}$ are orthogonal to
each other with lengths given by the corresponding stress coefficients. See
the figure aside for an illustration. It is important to note that the
existence of the Maxwell reciprocal diagram $G^{*}(p^{*})$ is equivalent to
non-zero self-stressability of $G(p)$.

\begin{overpic}[width=433.62pt]{fig/reciprocal} \end{overpic}

For $G(p)$ with self-stress $\omega$, let us denote the normal vector to the
face of the polyhedral lifting $L$ corresponding to a face $f\in F$ by
$\nu_{f}$ and let these vectors be “normalized” to $\nu_{f}=(*,*,-1)$. Then
from [5, Eqn. (4)] we know that

(1) $\nu_{f_{2}}-\nu_{f_{1}}=\omega_{ij}(p_{j}-p_{i})^{\perp}$

for neighbouring faces $f_{1},f_{2}\in F$ with common edge $p_{i}p_{j}$.

##  2\. Differential liftings in $\mathbb{R}^{2}$

In this section, we generalise the concept of the classical polyhedral lifting
(Definition 1.2). By Maxwell’s theorem (Theorem 1.3) the lifting exists
exactly for self-stressed frameworks over their embedding as planar straight-
line graph. Instead of considering the lifting itself we now switch to a
discrete differential of that lifting. It turns out that such a discrete
differential (which we call _differential lifting_ , cf. Definition 2.3)
exists also for frameworks which are not embedded as planar graphs as well as
for frameworks over non-planar graphs.

While an embedding of a planar graph constitutes a cell decomposition of
$\mathbb{R}^{2}$ into faces, a general realisation of a graph in
$\mathbb{R}^{2}$ might have crossing edges and divides the plane into cells
which we call _chambers_.

######  Definition 2.1.

For a framework $G(p)$, a _chamber_ is a connected component of the complement
of $G(p)$ to $\mathbb{R}^{2}$. Two different chambers are neighbours if they
share an edge (see Figure 2 left) or if they share a non-empty open sub-
segment of an edge (see Figure 2 right).

Note that as the framework $G(p)$ is finite, there is always an unbounded
chamber $C_{\infty}$, the one “around” the framework.

A key idea in our paper is to relate differential $1$-forms to the chambers of
the framework. For that we identify vectors $v=(v_{1},v_{2})$ with
differential $1$-forms $dv=v_{1}\,dx+v_{2}\,dy$ via linear combinations of
coordinate differentials $dx$ and $dy$. The space of differential $1$-forms is
denoted by $\Omega^{1}(\mathbb{R}^{2})$. For a given framework $G(p)$, we will
consider maps from the set of chambers to the space of constant $1$-forms on
$\mathbb{R}^{2}$. The space of chambers is canonically identified with the
elements of $\pi_{0}(\mathbb{R}^{2}\setminus G(p))$, the set of path
components. Once the map from the set of chambers to
$\pi_{0}(\mathbb{R}^{2}\setminus G(p))$ is fixed, we can say that the
corresponding $1$-forms are _associated_ with the chambers.

######  Definition 2.2.

Consider a framework $G(p)$ with a self-stress $\omega$. Let $C_{1}$ and
$C_{2}$ be two neighbouring chambers of $G(p)$ along $p_{i}p_{j}$ (i.e. they
are either sharing the same edge $p_{i}p_{j}$ or a non-empty open sub-segment
of it). The stress coefficient of $p_{i}p_{j}$ is $\omega_{ij}$. Let
$\alpha_{1}$ and $\alpha_{2}$ be two constant $1$-forms associated with
$C_{1}$ and $C_{2}$, respectively. We say that $\alpha_{1}$ and $\alpha_{2}$
satisfy the _neighbouring condition_ if

(2) $\alpha_{2}-\alpha_{1}={\rm
sign}(\det(\mathbf{n}_{12},p_{i}-p_{j}))\cdot\omega_{ij}\cdot
d(p_{i}{-}p_{j}),$

where $\mathbf{n}_{12}$ is the unit normal vector to $p_{i}-p_{j}$ pointing in
the direction from $C_{1}$ to $C_{2}$ (cf. Figure 2).

Note that the neighbouring condition is independent of the order of
$\alpha_{1}$ and $\alpha_{2}$.

\begin{overpic}[width=164.77771pt]{fig/chambers2} \put(27.0,61.0){$p_{i}$}
\put(27.0,-1.0){$p_{j}$} \put(17.0,30.0){$C_{1}$} \put(67.0,30.0){$C_{2}$}
\put(5.0,5.0){$C_{\infty}$} \put(39.0,35.0){$\mathbf{n}_{12}$}
\end{overpic}\begin{overpic}[width=164.77771pt]{fig/chambers1}
\put(27.0,61.0){$p_{i}$} \put(27.0,-1.0){$p_{j}$} \put(22.0,22.0){$C_{1}$}
\put(64.0,22.0){$C_{2}$} \put(48.0,42.0){$C_{3}$} \put(22.0,42.0){$C_{4}$}
\put(5.0,5.0){$C_{\infty}$} \put(39.0,23.0){$\mathbf{n}_{12}$} \end{overpic}

Figure 2. Two different general frameworks $G(p)$ and their chambers. Note
that two chambers are neighbours if they share an edge or a non-empty open
sub-segment of an edge. _Left:_ The framework consists of two neighbouring
faces, a triangle and a quadrilateral. The two neighbouring chambers $C_{1}$
and $C_{2}$ share the common edge $p_{i}p_{j}$. _Right:_ The framework has two
crossing edges which do not intersect in a vertex of the framework. The
pentagon is therefore divided into four chambers. Two neighbouring chambers,
e.g., $C_{1}$ and $C_{2}$, share a non-empty open sub-segment of the edge
$p_{i}p_{j}$.

The following definition of differential liftings is central for the present
paper. It generalises the concept of the classical polyhedral lifting
(Definition 1.2), in a sense, to a discrete differential of a lifting which
may or may not exist, as explained below.

######  Definition 2.3.

For the framework $G(p)$ and a self-stress $\omega$ of $G(p)$, we call a
mapping

$l_{\omega,p}:\pi_{0}(\mathbb{R}^{2}\setminus
G(p))\longrightarrow\Omega^{1}(\mathbb{R}^{2})$

a _differential lifting_ if it satisfies the _neighbouring condition_ for each
pair of neighbouring chambers and if it takes zero on the unbounded chamber of
$C_{\infty}\in\pi_{0}(\mathbb{R}^{2}\setminus G(p))$.

In Theorem 2.4, we show that a differential lifting as defined in Definition
2.3 exists and that it is unique. Note that the set of differential liftings
over a fixed framework $G(p)$ is a vector space with

$l_{\omega^{1},p}+l_{\omega^{2},p}=l_{\omega^{1}+\omega^{2},p}\qquad\text{and}\qquad\lambda
l_{\omega,p}=l_{\lambda\omega,p}.$

######  Theorem 2.4.

For every framework $G(p)$ with a self-stress $\omega$, there exists a unique
differential lifting $l_{\omega,p}$.

###### Proof.

Let us first assume that our framework $G(p)$ has no _crossing edges_.

By Definition 2.3, the value of the unbounded chamber $C_{\infty}$ is set to
be zero. Any other value of $l_{\omega,p}$ can be obtained by adding
differential forms via the neighbouring condition, Equation (2), along a path
of chambers starting at $C_{\infty}$, where a _path_ between two chambers is a
sequence of chambers that share a common edge. Our aim is to show that
different paths with the same starting and ending chambers do not give
different values at the last chamber.

Consider a path $\mathcal{C}$ of chambers $C_{1},\ldots,C_{k}$ with associated
differential forms denoted by $\alpha_{1},\ldots,\alpha_{k}$. Adding
differential forms along the path via Equation (2) yields

$\alpha_{k}=\alpha_{1}+\alpha_{\mathcal{C}},\qquad\hbox{where}\qquad\alpha_{\mathcal{C}}=\sum\limits_{r=2}^{k}(\alpha_{r}-\alpha_{r-1}),$

where $(\alpha_{r}-\alpha_{r-1})$ stands for the right hand side of Equation
(2).

For the reverse path $C_{k},\ldots,C_{1}$, the differential forms then satisfy
$\alpha_{1}=\alpha_{k}-\alpha_{\mathcal{C}}$.

Let us consider two paths $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ with the
same starting chamber $C_{s}$ and the same end chamber $C_{e}$. We will show
that these two (potentially different) paths give the same value at the last
chamber. Adding differential forms along the two paths yields

$\alpha^{1}_{e}=\alpha_{s}+\alpha_{\mathcal{C}_{1}}\qquad\text{and}\qquad\alpha^{2}_{e}=\alpha_{s}+\alpha_{\mathcal{C}_{2}}.$

Consequently, adding differential forms along the path $\mathcal{C}_{1}$
followed by the reversed path $\mathcal{C}_{2}$ (which combined is a _closed_
path as the starting chamber is the same as the end chamber) adds
$\alpha_{\mathcal{C}_{1}}-\alpha_{\mathcal{C}_{2}}$ to $\alpha_{s}$.
Therefore, the two paths $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ imply equal
values $\alpha^{1}_{e}=\alpha^{2}_{e}$ at $C_{e}$ if and only if the closed
path above just adds zero to $\alpha_{s}$.

The uniqueness of the definition of $l_{\omega,p}$ is therefore equivalent to
showing that adding differential forms via Equation (2), along closed paths,
adds zero to the starting chamber. We start with two elementary cycles before
we cover a general closed path.

_Case 1_ : Let us consider the closed path of chambers
$C_{1},C_{2},C_{3}=C_{1}$. This path corresponds to passing through an edge
and going back. Let us assume $l_{\omega,p}(C_{1})=\alpha_{1}$. Adding the
values along this path yields first (cf. Figure 2)

$\alpha_{2}=\alpha_{1}+\omega_{ij}\cdot{\rm
sign}(\det(\mathbf{n}_{12},p_{i}-p_{j}))\cdot d(p_{i}{-}p_{j}),$

and further

$\alpha_{3}=\alpha_{2}+\omega_{ij}\cdot{\rm
sign}(\det(\mathbf{n}_{23},p_{i}-p_{j}))\cdot d(p_{i}{-}p_{j}).$

since $\mathbf{n}_{23}=\mathbf{n}_{21}=-\mathbf{n}_{12}$, inserting these
values into the above formulas implies $\alpha_{3}=\alpha_{1}$. Consequently
nothing is added to the starting chamber.

_Case 2_ : Let us consider a closed path around a point $p_{i}$ in
counterclockwise direction, $C_{1},\ldots,C_{k}=C_{1}$. Recursively inserting
Equation (2) yields

$\alpha_{k}=\alpha_{1}+\sum_{j=1}^{k}\omega_{ij}\cdot{\rm
sign}(\det(\mathbf{n}_{j,j+1},p_{i}-p_{j}))\cdot
d(p_{i}{-}p_{j})=\alpha_{1}+d\Big{(}\sum_{j=1}^{k}\omega_{ij}(p_{i}-p_{j})\Big{)}=\alpha_{1},$

where we take indices modulo $k$, i.e. $\mathbf{n}_{k,k+1}=\mathbf{n}_{k,1}$.
The second equality holds true due to the linearity of the differential
operator, and because all the signs in the sum are positive (because of
counterclockwise orientation). The third equality holds by the equilibrium
condition for self-stressed frameworks at $p_{i}$.

_General case_ : Adding differential forms via (2) along any closed path adds
zero to the starting chamber since it can be decomposed into elementary closed
paths considered in Cases 1 and 2.

Let us now consider the case where the framework $G(p)$ with self-stress
$\omega$ has _crossing edges_ which we excluded so far. We can easily turn it
into a self-stressed framework without crossing edges. First, we add all
crossing points to the set of vertices and obtain a framework $G(\tilde{p})$.
Second, we modify the stress function $\omega$ to $\tilde{\omega}$ by the
following rule: Consider two neighbouring vertices
$\tilde{p}_{k}\tilde{p}_{l}$ forming an edge of $G(\tilde{p})$ which lies on
an “old” edge $p_{i}p_{j}$. Then the new stress coefficient
$\tilde{\omega}_{kl}$ is defined by

$\tilde{\omega}_{kl}:=\frac{\omega_{ij}}{\lambda_{kl}},$

where $\lambda_{kl}$ is the scaling factor between these two edges:
$\|p_{k}-p_{l}\|=\lambda_{kl}\|p_{i}-p_{j}\|$.

\begin{overpic}[width=433.62pt]{fig/newtensegrity.pdf}
\put(-11.0,51.0){$p_{i}{=}\tilde{p}_{i}$} \put(96.0,73.0){$p_{j}$}
\put(15.0,53.0){$\tilde{p}_{k}$} \put(35.0,58.0){$\tilde{p}_{l}$}
\put(62.0,65.0){$\tilde{p}_{m}$} \put(29.0,45.0){$C_{1}$}
\put(29.0,65.0){$C_{2}$} \end{overpic}

This is a self-stress since at every “new” vertex $\tilde{p}_{l}$ opposite
forces cancel:

$\tilde{\omega}_{kl}(\tilde{p}_{k}-\tilde{p}_{l})+\tilde{\omega}_{ml}(\tilde{p}_{m}-\tilde{p}_{l})=\frac{\omega_{ij}}{\lambda_{kl}}\lambda_{kl}(p_{i}-p_{j})+\frac{\omega_{ij}}{\lambda_{ml}}\lambda_{ml}(p_{j}-p_{i})=0.$

Analogously at “old” vertices $p_{i}$ the equilibrium condition still holds as
any “new” edge $\tilde{p}_{k}\tilde{p}_{i}$ contributes the same force:

$\tilde{\omega}_{ki}(\tilde{p}_{k}-\tilde{p}_{i})=\frac{\omega_{ij}}{\lambda_{ki}}\lambda_{ki}(p_{j}-p_{i})=\omega_{ij}(p_{j}-p_{i}).$

Now note that the neighbouring condition (2) along a “new” edge is the same as
before when considered as two neighbouring chambers along a non-empty open
sub-segment of an edge since

$\alpha_{2}-\alpha_{1}={\rm
sign}(\det(\mathbf{n}_{12},\tilde{p}_{k}-\tilde{p}_{l}))\cdot\tilde{\omega}_{kl}\cdot
d(\tilde{p}_{k}{-}\tilde{p}_{l})={\rm
sign}(\det(\mathbf{n}_{12},p_{i}-p_{j}))\cdot\omega_{ij}\cdot
d(p_{i}{-}p_{j}),$

which concludes the proof. ∎

\begin{overpic}[width=164.77771pt]{fig/k4-one} \put(33.0,47.0){$p_{1}$}
\put(83.0,44.0){$p_{2}$} \put(-4.0,92.0){$p_{3}$} \put(-4.0,-2.0){$p_{4}$}
\put(-4.0,50.0){\small$-1$} \put(34.0,81.0){\small$-1$}
\put(34.0,15.0){\small$-1$} \put(20.0,71.0){\small$3$}
\put(20.0,25.0){\small$3$} \put(57.0,46.0){\small$3$}
\put(60.0,87.0){\small\framebox{$0$}}
\put(6.0,47.0){\small\framebox{$-\sqrt{3}\,dy$}}
\put(29.0,37.0){\small\framebox{$\frac{3}{2}dx+\frac{\sqrt{3}}{2}dy$}}
\put(29.0,58.0){\small\framebox{$-\frac{3}{2}dx+\frac{\sqrt{3}}{2}dy$}}
\put(47.0,87.0){$C_{\infty}$} \put(31.0,69.0){$C_{1}$}
\put(11.0,60.0){$C_{2}$} \put(31.0,27.0){$C_{3}$}
\end{overpic}\begin{overpic}[width=173.44534pt]{fig/k4-two}
\put(96.0,55.0){$q_{1}$} \put(53.0,97.0){$q_{2}$} \put(-1.0,43.0){$q_{3}$}
\put(54.0,0.0){$q_{4}$} \put(23.0,79.0){\small$-1$}
\put(23.0,18.0){\small$-1$} \put(71.0,79.0){\small$-1$}
\put(71.0,18.0){\small$-1$} \put(52.0,26.0){\small$1$}
\put(30.0,52.0){\small$1$} \put(80.0,87.0){\small\framebox{$0$}}
\put(58.0,37.0){\small\framebox{$dx+dy$}}
\put(60.0,60.0){\small\framebox{$-dx+dy$}}
\put(20.0,60.0){\small\framebox{$-dx-dy$}}
\put(21.0,37.0){\small\framebox{$dx-dy$}} \put(66.0,91.0){$C_{\infty}$}
\put(59.0,71.0){$C_{1}$} \put(35.0,71.0){$C_{2}$} \put(35.0,24.0){$C_{3}$}
\put(61.0,24.0){$C_{4}$} \end{overpic}

Figure 3. Two frameworks corresponding to the combinatorial graph $K_{4}$.
With the vertices and stresses as detailed in Example 2.5 both frameworks are
self-stressed. The chambers are denoted by $C_{i}$ and the associated
$1$-forms are written with boxes. The _left_ framework $G(p)$ has a standard
lifting $L$ in the sense of Theorem 1.3 as well as a differential lifting
$l_{\omega,p}$. The framework $G(q)$ on the _right_ does not have a standard
lifting $L$ as in Theorem 1.3. It does, however, have a differential lifting
$l_{\omega,q}$.

######  Example 2.5.

Let us consider $K_{4}$, the smallest graph in the plane with a self-stress
for all framework realisations. We consider two distinct frameworks $K_{4}(p)$
and $K_{4}(q)$ as depicted in Figure 3. Here the framework in Figure 3 (left)
is planar whereas the framework in Figure 3 (right) is not. We have (cf.
Figure 3 left)

$\textstyle p_{1}=(0,0),\quad p_{2}=(1,0),\quad
p_{3}=(-\frac{1}{2},\frac{\sqrt{3}}{2}),\quad
p_{4}=(-\frac{1}{2},-\frac{\sqrt{3}}{2}),$

and stress coefficients $\omega_{12}=\omega_{13}=\omega_{14}=3$, and
$\omega_{23}=\omega_{34}=\omega_{24}=-1$. The chambers are denoted by

$C_{\infty}=p_{2}p_{3}p_{4},\quad C_{1}=p_{1}p_{2}p_{3},\quad
C_{2}=p_{1}p_{3}p_{4},\quad C_{3}=p_{1}p_{4}p_{2}.$

The $1$-form associated to the unbounded chamber $C_{\infty}$ is per
definition set to $0$. The associated $1$-forms corresponding to the other
chambers are

$\textstyle l_{\omega,p}(C_{1})=-\frac{3}{2}dx+\frac{\sqrt{3}}{2}dy,\quad
l_{\omega,p}(C_{2})=-\sqrt{3}\,dy,\quad
l_{\omega,p}(C_{3})=\frac{3}{2}dx+\frac{\sqrt{3}}{2}dy.$

For the non-planar realisation of $K_{4}$ (Figure 3 right) we have

$q_{1}=(1,0),\quad q_{2}=(0,1),\quad q_{3}=(-1,0),\quad q_{4}=(0,-1),$

where the edges $q_{1}q_{3}$ and $q_{2}q_{4}$ cross each other at the origin
denoted by $O=(0,0)$. The stress coefficients are
$\omega_{12}=\omega_{23}=\omega_{34}=\omega_{14}=-1$, and
$\omega_{13}=\omega_{24}=1$. The chambers are denoted by

$C_{\infty}=q_{1}q_{2}q_{3}q_{4},\quad C_{1}=Oq_{1}q_{2},\quad
C_{2}=Oq_{2}q_{3},\quad C_{3}=Oq_{3}q_{4},\quad C_{4}=Oq_{1}q_{4}.$

The $1$-form associated to the unbounded chamber $C_{\infty}$ is per
definition set to $0$. The associated $1$-forms corresponding to the other
chambers are

$l_{\omega,q}(C_{1})=dx-dy,\quad l_{\omega,q}(C_{2})=-dx-dy,\quad
l_{\omega,q}(C_{3})=-dx+dy,\quad l_{\omega,q}(C_{4})=dx+dy.$

The framework $G(q)$ in Figure 3 (right) does not have a standard lifting as
in Theorem 1.3, but as we showed, it has a differential lifting
$l_{\omega,q}$.

######  Example 2.6.

Another classical example is a framework with the combinatorics of a
triangular prism as depicted in Figure 4. The six vertices are

$p_{1}=(-1,-1),\quad p_{2}=(3,-1),\quad p_{3}=(3,1),\quad p_{4}=(-1,1),\quad
p_{5}=(0,0),\quad p_{6}=(2,0),$

and the stress coefficients are $\omega_{12}=\omega_{34}=-1$,
$\omega_{23}=\omega_{14}=-2$, and
$\omega_{15}=\omega_{45}=\omega_{26}=\omega_{36}=4$. The chambers are denoted
by

$C_{\infty}=p_{1}p_{2}p_{3}p_{4},\quad C_{1}=p_{1}p_{2}p_{6}p_{5},\quad
C_{2}=p_{2}p_{3}p_{6},\quad C_{3}=p_{3}p_{4}p_{5}p_{6},\quad
C_{4}=p_{1}p_{4}p_{5}.$

The $1$-form associated to the unbounded chamber $C_{\infty}$ is per
definition set to $0$. The associated $1$-forms corresponding to the other
chambers are

$l_{\omega,p}(C_{1})=-4\,dx,\quad l_{\omega,p}(C_{2})=4\,dy,\quad
l_{\omega,p}(C_{3})=4\,dx,\quad l_{\omega,p}(C_{4})=-4\,dy.$

\begin{overpic}[width=203.80193pt]{fig/prism} \put(-5.0,1.0){$p_{1}$}
\put(100.0,1.0){$p_{2}$} \put(100.0,49.0){$p_{3}$} \put(-5.0,49.0){$p_{4}$}
\put(26.0,22.0){$p_{5}$} \put(70.0,22.0){$p_{6}$}
\put(-6.0,25.0){\small\framebox{$0$}}
\put(6.0,25.0){\small\framebox{$-4\,dy$}}
\put(84.0,25.0){\small\framebox{$4\,dy$}}
\put(45.0,15.0){\small\framebox{$-4\,dx$}}
\put(45.0,35.0){\small\framebox{$4\,dx$}} \put(-8.0,14.0){$C_{\infty}$}
\put(49.0,7.0){$C_{1}$} \put(90.0,15.0){$C_{2}$} \put(47.0,43.0){$C_{3}$}
\put(7.0,15.0){$C_{4}$} \end{overpic} Figure 4. A framework corresponding to a
combinatorial triangular prism. With the vertices and stress coefficients as
detailed in Example 2.6, the framework is self-stressed. The chambers are
denoted by $C_{i}$ and the associated $1$-forms are written with boxes. The
framework $G(p)$ has a standard lifting $L$ in the sense of Theorem 1.3 and it
has a differential lifting denoted by $l_{\omega,p}$.

The relation between polygonal liftings and differential liftings on self-
stressed planar frameworks corresponding to planar graphs is given in the
following proposition.

######  Proposition 2.7.

Let $(G(p),\omega)$ be a self-stressed planar framework corresponding to a
planar graph. Let furthermore $L$ denote a polyhedral lifting as in Definition
1.2 and let $l_{\omega,p}$ be a differential lifting. Consider the
“normalized” normal vector $\nu_{C}=(*,*,-1)$ of the polyhedral lifting
corresponding to the chamber $C$ and let $\alpha_{C}=l_{\omega,p}(C)$ be the
associated $1$-form. Then

$\star\alpha_{C}=\pm\nu_{C}\cdot\Big{(}\begin{smallmatrix}dx\\\ dy\\\
0\end{smallmatrix}\Big{)},$

where $\star$ denotes the _Hodge star operator_ , i.e.
$\star(a\,dx+b\,dy)=-b\,dx+a\,dy$.

###### Proof.

This follows from Equations (1) and (2). ∎

The condition of the graph being planar in Theorem 1.3 is not needed when we
consider differential liftings instead of polyhedral liftings.

######  Theorem 2.8.

Let $G$ be any finite simple graph and $G(p)$ be a framework in
$\mathbb{R}^{2}$. The mapping $(G(p),\omega)\mapsto l_{\omega,p}$ is a
bijection between the space of self-stresses and the space of differential
liftings.

###### Proof.

First note that the space of self-stresses and the space of differential
liftings are vector spaces. The mapping is linear as well as surjective since
$l_{\omega,p}$ is well defined by Theorem 2.4.

Let two self-stresses have the same differential lifting. Then their
difference is lifted to the “zero lifting”, which, by Equation (2), is
possible only if the stress coefficients on all edges are zero. ∎

##  3\. Differential liftings in $\mathbb{R}^{n}$

Consider a framework $G(p)$ in $\mathbb{R}^{n}$. For $n\geq 3$ the complement
$\mathbb{R}^{n}\setminus G(p)$ does not decompose into disconnected chambers
but rather consists of one connected component. Consequently we replace the
notion of chambers by the notion of equivalence classes of oriented loops in
the complement of $G(p)$ to $\mathbb{R}^{n}$. Here the base point of the loops
can be arbitrarily chosen to be any point of the complement.

Similarly to the differential liftings of Section 2 (which are maps from
$\pi_{0}(\mathbb{R}^{2}\setminus G(p))$ to $\Omega^{1}(\mathbb{R}^{2})$) we
set differential liftings in $\mathbb{R}^{n}$ to be special maps of the
fundamental group $\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$ to
$\Omega^{1}(\mathbb{R}^{n})$ (Definition 3.6). In our setting homotopic loops
will map to the same associated $1$-form.

Given a vector $v=(v_{1},\ldots,v_{n})$, we identify it with the differential
$1$-form $dv=v_{1}\,dx_{1}+\cdots+v_{n}\,dx_{n}$ via linear combinations of
coordinate differentials.

For a given framework, we consider maps from the fundamental group
$\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$ to the space of constant $1$-forms
in $\mathbb{R}^{n}$. Once the map is fixed, we can say that the corresponding
$1$-forms are _associated_ with the homotopy classes of loops.

The number of how many times two disjoint manifolds are intertwined is called
_linking number_. Let us recall its definition from the original paper by
Gauss [14, p. 605]. In the literature, such as [23, p. 53], one often also
finds another convention for the sign of this definition.

######  Definition 3.1.

Consider two disjoint oriented manifolds
$M_{1},M_{2}\subset\mathbb{R}^{m_{1}+m_{2}+1}$ of respective dimensions
$m_{1}$ and $m_{2}$. Then the _linking map_ is the map

$\lambda:M_{1}\times M_{2}\longrightarrow S^{m_{1}+m_{2}}$

defined by $\lambda(x,y)=-(x-y)/\|x-y\|$. If $M_{1}$ and $M_{2}$ are compact,
oriented and without boundary, then the degree of $\lambda$ is called the
_linking number_ ${\rm lk}(M_{1},M_{2})$.

######  Remark 3.2.

The orientation of the sphere $S^{m_{1}+m_{2}}$ is important for the
definition of the linking number. We fixed the orientation in such a way that
at point $(1,0,\ldots,0)\in S^{m_{1}+m_{2}}$ it coincides with the orientation
of the basis $(e_{2},\ldots,e_{m_{1}+m_{2}+1})$. In such settings the
orientations of Hopf links in $\mathbb{R}^{3}$ are as in Figure 5.

\begin{overpic}[width=216.81pt]{fig/hopf-links.jpg} \put(-2.0,22.0){$M_{1}$}
\put(38.0,22.0){$M_{2}$} \put(56.0,22.0){$M_{1}$} \put(96.0,22.0){$M_{2}$}
\put(-3.0,7.0){ \leavevmode\hbox to35.61pt{\vbox
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(42.0,7.0){
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(55.0,7.0){
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(101.0,7.0){
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to27.72pt{\pgfpicture\makeatletter\hbox{\hskip-24.62234pt\lower-13.85764pt\hbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.46947}{-0.88295}{0.88295}{-0.46947}{27.57298pt}{-8.74861pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \end{overpic} Figure
5. A negatively (_left_) and a positively (_right_) oriented Hopf link of two
oriented circles in $\mathbb{R}^{2}$.

######  Remark 3.3.

Below, one of the manifolds will be a piecewise linear finite polyhedral
$m$-sphere. In this case we consider the linking number of the limit of
approximations by differentiable spheres in $\varepsilon$ neighbourhoods of
the piecewise linear sphere in the continuous topology while $\varepsilon$
tends to zero.

The linking number is an integer which is homotopy invariant and can be
computed with the so called multi-dimensional Gauss linking integral (see,
e.g., [34]). Coming back to our frameworks $G(p)$, for a simple cycle $\delta$
in $G(p)$ and $\Gamma\in\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$ we set

${\rm lk}(\Gamma,\delta)={\rm lk}(\gamma,\delta)$

with $\gamma$ being any element of $\Gamma$. The linking number of two
manifolds is well defined since it is invariant under homotopies and hence it
does not depend on the choice of $\gamma\in\Gamma$.

\begin{overpic}[width=203.80193pt]{fig/loops} \put(-2.0,29.0){$p_{i}$}
\put(97.0,50.0){$p_{j}$} \put(60.0,40.0){$e$} \put(24.0,22.0){$\gamma_{1}$}
\put(41.0,22.0){$\gamma_{2}$} \end{overpic} Figure 6. Two loops
$\gamma_{1}\in\Gamma_{1}$ and $\gamma_{2}\in\Gamma_{2}$ from neighbouring
homotopy classes $\Gamma_{1}$ and $\Gamma_{2}$ along the edge $e=p_{i}p_{j}$.
There is a homotopy between $\gamma_{1}$ and $\gamma_{2}$ which transversally
crosses $e$ only once.

We say that two homotopy classes
$\Gamma_{1},\Gamma_{2}\in\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$ are
_neighbours at an edge $e$_ if there is a homotopy between loops
$\gamma_{1}\in\Gamma_{1}$ and $\gamma_{2}\in\Gamma_{2}$ in $\mathbb{R}^{n}$
crossing the framework along $e$ transversally only once. See Figure 6 for an
illustration of loops from neighbouring homotopy classes. For each edge
$e=p_{i}p_{j}$ there is a homotopy class which is a neighbour to the neutral
element of $\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$ along $e$ (see Figure 7).
We call it an _elementary_ homotopy class.

Consider a homotopy class $\Gamma\in\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$
and let $\Theta$ be an elementary homotopy class neighbouring the neutral
element along $e=p_{i}p_{j}$. Then $\Gamma\pm\Theta$ is a neighbour of
$\Gamma$ at $p_{i}p_{j}$.

Any homotopy class can be unwound by adding or subtracting elementary loops to
finally become the neutral element. Consequently, between any two homotopy
classes of $\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$ there is a _path_ of
neighbouring homotopy classes. Unwinding to the neutral element and winding up
to the target class would be such a path.

\begin{overpic}[width=130.08731pt]{fig/fundamental-group}
\put(25.0,19.0){$\gamma$} \put(84.0,11.0){$\theta$} \put(78.0,44.0){$e$}
\end{overpic}\begin{overpic}[width=130.08731pt]{fig/fundamental-group1}
\put(25.0,19.0){$\gamma+\theta$} \put(78.0,44.0){$e$}
\end{overpic}\begin{overpic}[width=130.08731pt]{fig/fundamental-group2}
\put(25.0,19.0){$\gamma-\theta$} \put(77.0,46.0){$e$} \end{overpic}

Figure 7. A framework $G(p)$ in $\mathbb{R}^{3}$ and loops from
$\pi_{1}(\mathbb{R}^{3}\setminus G(P))$. _Left:_ The homotopy class $\Gamma$
is given by a representing loop $\gamma\in\Gamma$ and the elementary homotopy
class $\Theta$ is given by an elementary loop $\theta\in\Theta$. The homotopy
class $\theta$ is a neighbour to the neutral element at edge $e$. _Centre and
Right:_ The sums $\Gamma\pm\Theta$ are both neighbours to $\Gamma$ at edge
$e$.

######  Definition 3.4.

Consider a framework $G(p)$ with a self-stress $\omega$. Let $\Gamma_{1}$ and
$\Gamma_{2}$ be two different homotopy classes of loops that are neighbours at
an edge $p_{i}p_{j}$ with stress coefficient $\omega_{ij}$. Fix a simple cycle
$\gamma$ in $G(p)$ which includes the edge $p_{i}p_{j}$ with orientation from
$p_{i}$ to $p_{j}$. Let $\alpha_{1},\alpha_{2}\in\Omega^{1}(\mathbb{R}^{n})$
be two constant $1$-forms associated with $\Gamma_{1}$ and $\Gamma_{2}$. We
say that $\alpha_{1}$ and $\alpha_{2}$ satisfy the _neighbouring condition_ if

(3) $\alpha_{2}-\alpha_{1}=\big{(}{\rm lk}(\Gamma_{1},\gamma)-{\rm
lk}(\Gamma_{2},\gamma)\big{)}\cdot\omega_{ij}\cdot d(p_{i}{-}p_{j}).$

This definition is neither dependent on the choice of cycle $\gamma$ nor on
the orientation of the edge $p_{i}p_{j}$: the choice of the simple cycle in
Definition 3.4 does not affect Equation (3) since $\big{(}{{\rm
lk}}(\gamma,\Gamma_{1})-{\rm lk}(\gamma,\Gamma_{2})\big{)}$ stays the same due
to the orientation choice. Furthermore, the choice of the orientation of the
edge $p_{i}p_{j}$ does not affect the right hand side of Equation (3) since
the linking number changes sign and $p_{j}-p_{i}=-(p_{i}-p_{j})$.

######  Remark 3.5.

Note that since $\Gamma_{1}$ and $\Gamma_{2}$ are neighbours, we have

${\rm lk}(\Gamma_{1},\gamma)-{\rm lk}(\Gamma_{2},\gamma)=\pm 1.$

######  Definition 3.6.

For the framework $G(p)$ and a self-stress $\omega$ of $G(p)$, we call a
mapping

$l_{\omega,p}:\pi_{n-2}(\mathbb{R}^{n}\setminus
G(p))\longrightarrow\Omega^{1}(\mathbb{R}^{n})$

a _differential lifting_ , if $l_{\omega,p}(\Gamma)$ is a constant $1$-form
for all $\Gamma\in\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$, if it satisfies
the _neighbouring condition_ for each neighbouring homotopy classes, and if it
takes zero on the neutral element
$\Gamma_{\infty}\in\pi_{n-2}(\mathbb{R}^{2}\setminus G(p))$.

The following theorem is the higher dimensional analogue to Theorem 2.4 of the
$2$-dimensional case.

######  Theorem 3.7.

For every framework $G(p)$ with a self-stress $\omega$, there exists a unique
differential lifting $l_{\omega,p}$.

###### Proof.

For the simplicity of the exposition, we call the elements of
$\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$ chambers, similarly to the case
$n=2$ considered in the previous section.

Existence: Let us say that a _path_ between two chambers is a sequence of
chambers that are neighbouring. The path is _closed_ if the starting chamber
is the same as the end chamber.

The _monodromy_ of $\Omega^{1}(\mathbb{R}^{n})$ associated to a closed path is
the linear mapping defined by the summation of all forms on the right hand
side of Equation (3).

Let us show that all monodromies are zero.

_Case 1_ : Let us consider monodromies for the closed path
$C_{1},C_{2},C_{1}$. They correspond to passing through an edge and going
back, which will result in

$\alpha\mapsto\alpha+\big{(}{\rm lk}(\Gamma_{1},\gamma)-{\rm
lk}(\Gamma_{2},\gamma)\big{)}\omega_{ij}\cdot d(p_{i}{-}p_{j})+\big{(}{\rm
lk}(\Gamma_{2},\gamma)-{\rm lk}(\Gamma_{1},\gamma)\big{)}\omega_{ij}\cdot
d(p_{i}{-}p_{j})=\alpha.$

_Case 2_ : Here we need to extend the definition of a closed path around a
point $p_{i}$. Consider a small sphere $S^{n-1}$ centered at $p_{i}$ and let
us fix the marked point $O_{n-1}$ for the homotopy group so that it lies in
$S^{n-1}\setminus G(p)$. Let us consider a differential homotopy
$H:[0,1]\times S^{n-2}\to S^{n-1}$ such that

  * •

$H(\\{0\\}\times S^{n-2})=\\{O_{n-1}\\}$ and $H(\\{1\\}\times
S^{n-2})=\\{O_{n-1}\\}$;

  * •

$H([0,1]\times\\{O_{n-2}\\})=\\{O_{n-1}\\}$;

  * •

$H:(0,1)\times S^{n-2}\setminus\\{O_{n-2}\\}\to S^{n-1}\setminus\\{O_{n-1}\\}$
is a differentiable, non-singular bijection with positive Jacobian;

  * •

The pre-images of points $G(p)\cap S^{n-1}$ have pairwise different first
coordinates.

This homotopy induces the closed path between the chambers starting and ending
in the identity chamber. We say that this path has a _counterclockwise_
direction. If we reverse the first coordinate, then we have _clockwise_
direction.

Let us consider a closed path around a point $p_{i}$ in clockwise direction,
$C_{1},\ldots,C_{s},C_{1}$. Then the monodromy is

(4) $\alpha\mapsto\alpha+\sum_{i=1}^{s}\big{(}{\rm lk}(\Gamma_{i},\gamma)-{\rm
lk}(\Gamma_{i+1},\gamma)\big{)}\omega_{ij}\cdot
d(p_{i}{-}p_{j})=\alpha+d\Big{(}\sum_{i=1}^{s}\omega_{i,j}(p_{i}{-}p_{j})\Big{)}=\alpha.$

The first equality is true by the linearity of the differentials and since the
differences between all linking numbers are $+1$. The second equality holds by
the equilibrium condition at $p_{i}$.

_General case_ : The monodromy for any closed path is zero since it can be
decomposed into the elementary closed paths considered in Cases 1 and 2.

Therefore, if we have two paths $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ with
the same starting chamber $C_{s}$ and the same end chamber $C_{e}$, then the
corresponding paths through neighbours will lead to the same differential
form, since the closed path $\mathcal{C}_{2}$ followed by the reversed path
$\mathcal{C}_{1}$ has zero monodromy.

Uniqueness: By definition, the form corresponding to the unbounded element of
$\pi_{0}(\mathbb{R}^{n}\setminus G(p))$ (denoted by $C_{\infty}$) is zero.
Then each path from $C_{\infty}$ to $C$ uniquely determines the form for $C$
by Equation (2). ∎

######  Remark 3.8.

Let $\Gamma=\Gamma_{1}+\Gamma_{2}$. Then

$l_{\omega,p}(\Gamma)=l_{\omega,p}(\Gamma_{1})+l_{\omega,p}(\Gamma_{2}).$

Therefore, once we know the values of the differential liftings for a set of
generators of $\pi_{n-2}(\mathbb{R}^{n}\setminus G(p))$, we get all the values
for $l_{\omega,p}$ by the above additivity.

######  Theorem 3.9.

Let $G$ be any graph and $G(p)$ be a framework in $\mathbb{R}^{n}$. The
mapping $l_{\omega,p}$ is a bijection between the space of self-stresses and
the space of differential liftings.

###### Proof.

The proof is the same as the proof of Theorem 2.8. ∎

######  Remark 3.10.

Note that geometric realisability of self-stresses was studied in [17].

######  Example 3.11.

Consider a framework with graph $K_{5}$ in 3-space with the following
vertices:

$p_{1}=(0,0,0),\quad p_{2}=(1,0,0),\quad p_{3}=(0,1,0),\quad
p_{4}=(0,0,1),\quad p_{5}=(1,1,1).$

This framework has a one-dimensional space of self-stresses. Let $\omega$ be
one of these self-stresses defined by the matrix:

$\omega=\begin{bmatrix}\phantom{-}0&\phantom{-}2&\phantom{-}2&\phantom{-}2&-2\\\
\phantom{-}2&\phantom{-}0&-1&-1&\phantom{-}1\\\
\phantom{-}2&-1&\phantom{-}0&-1&\phantom{-}1\\\
\phantom{-}2&-1&-1&\phantom{-}0&\phantom{-}1\\\
-2&\phantom{-}1&\phantom{-}1&\phantom{-}1&\phantom{-}0\\\ \end{bmatrix},$

where the stress coefficient $\omega_{i,j}$ is in position $(i,j)$. Then the
neighbouring condition along the edge $p_{i}p_{j}$ is

$\alpha_{2}-\alpha_{1}=\big{(}{\rm lk}(\Gamma_{1},\gamma)-{\rm
lk}(\Gamma_{2},\gamma)\big{)}\cdot\beta_{i,j}.$

The collection of $1$-forms $\beta_{i,j}$ is presented by the following matrix
(we consider $(x,y,z)$ as coordinates of 3-space):

$\beta=\begin{bmatrix}0&-2dx&-2dy&-2dz&2dx+2dy+2dz\\\ 2dx&0&-dx+dy&-dx+dz&-dy-
dz\\\ 2dy&dx-dy&0&-dy+dz&-dx-dz\\\ 2dz&dx-dz&dy-dz&0&-dx-dy\\\
-2dx-2dy-2dz&dy+dz&dx+dz&dx+dy&0\\\ \end{bmatrix}.$

Consider the loops in Figure 8. The homotopy class $\Gamma_{1}$ represented by
$\gamma_{1}$ is a neighbour to the homotopy class of trivial loops at the edge
$p_{1}p_{2}$, and hence

$l_{\omega,p}(\Gamma_{1})=-2dx.$

In order to get from the homotopy class of trivial loops to the homotopy class
$\Gamma_{2}$ one may cross first the edge $p_{2}p_{4}$ and then the edge
$p_{2}p_{5}$. So we have:

$l_{\omega,p}(\Gamma_{2})=(-dx+dz)+(-dy-dz)=-dx-dy.$

\begin{overpic}[width=130.08731pt]{fig/k5.jpg} \put(10.0,0.0){$p_{1}$}
\put(93.0,5.0){$p_{2}$} \put(-4.0,22.0){$p_{3}$}
\put(10.0,90.0){\contour{white}{$p_{4}$}}
\put(70.0,97.0){\contour{white}{$p_{5}$}} \end{overpic}

\begin{overpic}[width=130.08731pt]{fig/k5-with-loops.jpg}
\put(10.0,0.0){$p_{1}$} \put(90.0,8.0){$p_{2}$} \put(-4.0,22.0){$p_{3}$}
\put(10.0,88.0){\contour{white}{$p_{4}$}}
\put(66.0,93.0){\contour{white}{$p_{5}$}} \put(51.0,0.0){$\gamma_{1}$}
\put(93.0,30.0){$\gamma_{2}$} \end{overpic}

Figure 8. Loops linked with $K_{5}$.

######  Remark 3.12.

Definition 3.4 coincides with Definition 2.2 in the case when $n=2$. Here we
consider $\pi_{0}$, which coincides with connected components. In this case,
$\Gamma_{1}$ and $\Gamma_{2}$ are represented by single points $x_{1}$ and
$x_{2}$ in the corresponding chambers. Then we have

${\rm lk}(x_{1},\gamma)-{\rm lk}(x_{2},\gamma)={\rm wind}(\gamma,x_{1})-{\rm
wind}(\gamma,x_{2})={\rm sign}(\det(\mathbf{n}_{12},p_{i}-p_{j})),$

where $\mathbf{n}_{12}$ is the normal of the vector $p_{i}-p_{j}$ in the
direction from the first to the second chamber and where ‘${\rm wind}$’
denotes the winding number. The first equation holds because the linking
numbers coincide with the winding numbers. The second equation is a general
property of the winding number.

##  4\. Differential liftings of polytopal embeddings of complexes

In this section we will generalise differential liftings to higher dimensional
frameworks. We will investigate structures which are polyhedral CW complexes
where all faces in which forces appear are of the same dimension. We introduce
$m$-frameworks and their equilibrium stresses.

###  4.1. The $m$-frameworks

Let us recall a general definition of $m$-frameworks from [18]. Our model for
self-stressable $m$-frameworks in this section will be based on the following
structure. The term _plane_ will refer to an affine subspace in
$\mathbb{R}^{n}$. In the following let $n>m\geq 1$.

######  Definition 4.1.

An _$m$ -framework_ $\mathcal{F}=(E,F,I,\mathbf{n})$ consists of $E$, a
collection of $(m{-}1)$-dimensional planes in $\mathbb{R}^{n}$; $F$, a
collection of $m$-dimensional planes in $\mathbb{R}^{n}$; a subset
$I\subset\\{(e,f)\in E\times F\mid e\subset f\\}$; a function $\mathbf{n}$
assigning to each pair $(e,f)\in I$, a unit vector $\mathbf{n}(e,f)$ which is
contained in $f$ and which is orthogonal to $e$. We call planes from $F$
_faces_ and planes from $E$ _edges_.

######  Definition 4.2.

Let $\mathcal{F}=(E,F,I,\mathbf{n})$ be an $m$-framework. A _force-load_
$\omega$ on $\mathcal{F}$ is any function $\omega:F\to\mathbb{R}$. A framework
$\mathcal{F}$ together with a force-load $\omega$ is said to be in
_equilibrium_ if for every $e\in E$ we have

$\sum_{(e,f)\in I}\omega(f)\mathbf{n}(e,f)=0.$

Such a force-load is called an _equilibrium force-load_ for $\mathcal{F}$.

Note that there is a difference between force-loads and stresses for graphs.
The latter is related the length of the edges whereas force-loads are not. For
now stresses are not defined as faces of $m$-frameworks are planes and do not
have specified shapes or magnitudes. After introducing polytopal complexes
below we could consider stresses as well.

######  Definition 4.3.

An $m$-framework is said to be _self-stressable_ or a _tensegrity_ if there
exists a non-zero equilibrium force-load on it.

###  4.2. Polytopal embeddings of complexes

An _$m$ -dimensional polytope_ is the convex hull of a finite set of points
spanning $\mathbb{R}^{m}$. The boundary is piecewise linear so we have a
natural structure of faces. The faces of dimension $m{-}1$ are called
_facets_.

######  Definition 4.4.

Consider a finite collection $F$ of $m$-dimensional polytopes in
$\mathbb{R}^{n}$ whose intersection is either a facet or a $q$-dimensional
polytope with $q\leq m-2$. Let $E$ be the collection of all facets of the
polytopes in $F$ (here we identify the same facets of distinct polytopes). We
also require that the planes of any two facets in $E$ are distinct. Then the
pair $\mathcal{C}=(F,E)$ is called an _$m$ -dimensional polytopal complex_ in
$\mathbb{R}^{n}$.

######  Definition 4.5.

The $m$-framework $\mathcal{F}_{C}=(E_{C},F_{C},I_{C},\mathbf{n})$
_associated_ with an $m$-dimensional polytopal complex $C$ is defined by

  * •

$E_{C}$, the collection of $(m{-}1)$-dimensional planes spanned by facets of
$E$;

  * •

$F_{C}$, the collection of $m$-dimensional planes spanned by polytopes of $F$;

  * •

$I_{C}=\\{(e,f)\in E_{C}\times F_{C}\mid\text{the facet}\ e\ \text{is a facet
of polytope}\ f\\}$;

  * •

a function $\mathbf{n}$ assigning to each pair $(e,f)\in I_{C}$ a unit vector
$\mathbf{n}(e,f)$ which is contained in $f$, is orthogonal to $e$, and points
to the interior of the polytope.

######  Example 4.6.

Let us consider a very broad family of examples arising from parallel meshes
in $\mathbb{R}^{3}$. We say that two polyhedral surfaces $P_{1}$ and $P_{2}$
are _parallel_ if $P_{1}$ and $P_{2}$ have the same combinatorics and such
that corresponding edgwes are parallel (consequently also corresponding faces
are parallel). Let us connect the corresponding parallel edges of such a pair
of meshes. Then we get trivalent $2$-dimensional polytopal complexes (see an
example in Figure 9). For further details on parallel surfaces and related
discrete differential geometry we refer to [3] and [19].

\begin{overpic}[width=216.81pt]{fig/offsetmesh.jpg} \end{overpic} Figure 9. An
example of a pair of parallel meshes.

###  4.3. Differential liftings of $m$-frameworks

Consider an $m$-dimensional polytopal complex $C$. While for graphs we worked
with $1$-forms, we now work with $m$-forms. As before, we will study
fundamental groups $\pi_{n-m-1}(\mathbb{R}^{n}\setminus C)$. Again we can
choose the base point to be any point of the complement. Note that homotopic
$(n{-}m{-}1)$-spheres will be associated with the same $m$-form.

For an oriented $m$-dimensional polytope $f$ we set the differential form

$\alpha_{f}=de_{1}\wedge\ldots\wedge de_{m},$

where $e_{1},\ldots,e_{m}$ is an orthonormal basis of the span of $f$ with
positive orientation.

For a given $m$-dimensional polytopal complex $C$, we consider maps from the
fundamental group $\pi_{n-m-1}(\mathbb{R}^{n}\setminus C)$ to the space of
constant $m$-forms in $\mathbb{R}^{n}$. Once the map is fixed, we say that the
corresponding $m$-forms are _associated_ with the homotopy classes of spheres.

We say that two homotopy classes of spheres in $\mathbb{R}^{n}\setminus C$ are
_neighbours at a facet_ $e$ if there is a homotopy of spheres in
$\mathbb{R}^{n}$ crossing the framework along $e$ only once (transversally).

An _$m$ -cycle_ in $C$ is an oriented sub-complex of $C$ homeomorphic to the
unit $m$-sphere.

######  Definition 4.7.

Consider an $m$-dimensional polytopal complex $C$ with a non-zero equilibrium
force-load $\omega$ on the associated $m$-framework $\mathcal{F}_{C}$. Let
$\Gamma_{1}$ and $\Gamma_{2}$ be two different homotopy classes of spheres
that are neighbours at an oriented facet $f$ with stress coefficient
$\omega_{f}$. Fix a simple $m$-cycle $\gamma$ in $C$ which includes the facet
$f$ with orientation of $f$. Let $\alpha_{1}$ and $\alpha_{2}$ be two constant
$m$-forms associated with $\Gamma_{1}$ and $\Gamma_{2}$. We say that
$\alpha_{1}$ and $\alpha_{2}$ satisfy the _neighbouring condition_ if

(5) $\alpha_{2}-\alpha_{1}=({\rm lk}(\Gamma_{1},\gamma)-{\rm
lk}(\Gamma_{2},\gamma))\cdot\omega_{f}\cdot\alpha_{f}.$

######  Remark 4.8.

Note that since $\Gamma_{1}$ and $\Gamma_{2}$ are neighbours, we have

${\rm lk}(\Gamma_{1},\gamma)-{\rm lk}(\Gamma_{2},\gamma)=\pm 1.$

######  Remark 4.9.

The choice of the simple $m$-cycle in Definition 4.7 does not affect Equation
(5) since $({\rm lk}(\gamma,\Gamma_{1})-{\rm lk}(\gamma,\Gamma_{2}))$ does not
change due to the orientation choice.

Note also that the choice of the orientation of the facet $f$ does not affect
the right hand side of Equation (5) since the linking number and the form
$\alpha_{f}$ simultaneously change their signs.

######  Definition 4.10.

Consider an $m$-dimensional polytopal complex $C$ with a non-zero force-load
$\omega$ on the associated $m$-framework $\mathcal{F}_{C}$. The _differential
lifting_ $l_{\omega,C}$ is the $m$-form which

  * •

is constant on the connected components of the complement of $C$ to
$\mathbb{R}^{n}$;

  * •

satisfies the neighbouring condition for all neighbouring homotopy classes of
spheres;

  * •

is zero on the homotopy class of spheres that includes the constant map.

Denote by $l_{\omega,C}(\Gamma)$ the form corresponding to
$\Gamma\in\pi_{n-m-1}(\mathbb{R}^{n}\setminus C)$.

######  Theorem 4.11.

For every $m$-framework $\mathcal{F}_{C}$ with a force-load $\omega$, there
exists a unique differential lifting of $C$.

###### Proof.

For the simplicity of the exposition, we call the elements of
$\pi_{n-m-1}(\mathbb{R}^{n}\setminus G(p))$ the _chambers_ , similarly to the
case $m=1$ considered in the previous section.

Existence: Let us say that a _path_ between two chambers is a sequence of
chambers that are neighbouring. The path is _closed_ if the starting chamber
is the same as the end chamber.

The _monodromy_ of $\Omega^{m}(\mathbb{R}^{n})$ associated to a closed path is
the linear mapping defined by the summation of all forms that are second
summands on the right hand side of Equation (5).

Let us show that all monodromies are zero.

_Case 1_ : Let us consider monodromies for the closed path
$\Gamma_{1},\Gamma_{2},\Gamma_{1}$. They correspond to passing through a
polytope $f$ and going back, which will result in

$\alpha\mapsto\alpha+\big{(}{\rm lk}(\Gamma_{1},\gamma)-{\rm
lk}(\Gamma_{2},\gamma)\big{)}\omega_{ij}\cdot\alpha_{f}+\big{(}{\rm
lk}(\Gamma_{2},\gamma)-{\rm
lk}(\Gamma_{1},\gamma)\big{)}\omega_{ij}\cdot\alpha_{f}=\alpha.$

_Case 2_ : Here we need to extend the definition of a closed path around a
facet $e$ in clockwise direction. It is done similarly to the case $m=1$ by
restricting our $(n{-}m{-}1)$-spheres to the orthogonal complement to $e$,
which is $(n{-}m{+}1)$-dimensional.

Let us consider a closed path around a facet $e$ in clockwise direction,
$\Gamma_{1},\ldots,\Gamma_{s},\Gamma_{1}$. Then the monodromy is

$\alpha\mapsto\alpha+\sum_{i=1}^{s}\big{(}{\rm lk}(\Gamma_{i},\gamma)-{\rm
lk}(\Gamma_{i+1},\gamma)\big{)}\omega_{ij}\cdot\alpha_{e}\wedge\mathbf{n}(e,f)=\alpha+\alpha_{e}\wedge
d\Big{(}\sum_{i=1}^{s}\omega_{i,j}\mathbf{n}(e,f)\Big{)}=\alpha.$

The first equality is true by the linearity of the differentials and since the
differences between all linking numbers are $+1$. The second equality holds by
the equilibrium condition at $e$.

_General case_ : The monodromy for any closed path is zero since it is
decomposable into the elementary closed paths considered in Cases 1 and 2.

Therefore, if we have two paths $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ with
the same starting chamber $\Gamma_{s}$ and the same end chamber $\Gamma_{e}$,
then the corresponding paths through neighbours will lead to the same
differential form, since the closed path $\mathcal{C}_{2}$ followed by the
reversed path $\mathcal{C}_{1}$ has zero monodromy.

Uniqueness: This follows from exactly the same argument as in the proof of
Theorem 4.11. ∎

######  Remark 4.12.

Consider $\Gamma,\Gamma_{1},\Gamma_{2}\in\pi_{n-m-1}(\mathbb{R}^{n}\setminus
C)$ satisfying the identity $\Gamma=\Gamma_{1}+\Gamma_{2}$. Then

$l_{\omega,C}(\Gamma)=l_{\omega,C}(\Gamma_{1})+l_{\omega,C}(\Gamma_{2}).$

Therefore, once we know the values of the differential liftings for a set of
generators of $\pi_{n-m-1}(\mathbb{R}^{n}\setminus C)$, we get all the values
for $l_{\omega,C}$ by the above additivity.

######  Theorem 4.13.

Let $C$ be an $m$-dimensional polytopal complex. The mapping $(C,\omega)\to
l_{\omega,C}$ is a bijection between the space of force-loads and the space of
differential liftings.

###  4.4. Stresses and force-loads, and relation to the case $m=1$

The general situation we considered in this section has one minor difference
from the settings of the previous two sections. Here, we considered force-
loads rather than stresses. As we know, the equilibrium conditions for
stresses are

$\sum\limits_{\\{j\mid j\neq i\\}}\omega_{ij}(p_{i}-p_{j})=0,$

while the equilibrium conditions for force-loads are normalised as

$\sum\limits_{\\{j\mid j\neq
i\\}}\omega_{ij}\frac{p_{i}-p_{j}}{\|p_{i}-p_{j}\|}=0.$

For polytopes, we could also consider the corresponding stresses defined by
the equilibrium condition

$\sum_{(e,f)\in I}\textrm{Vol}(f_{*})\omega(f)\mathbf{n}(e,f)=0,$

where $f_{*}$ is the facet corresponding to the $m$-plane $f$.

Now let us recall the Equation (3):

$\alpha_{2}=\alpha_{1}+({\rm lk}(\Gamma_{1},\gamma)-{\rm
lk}(\Gamma_{2},\gamma))\cdot\omega_{ij}\cdot d(p_{i}{-}p_{j}).$

This equation coincides with the non-normalised version of Equation (5):

$\alpha_{2}=\alpha_{1}+\textrm{Vol}(f_{*})\big{(}({\rm
lk}(\Gamma_{1},\gamma)-{\rm
lk}(\Gamma_{2},\gamma))\cdot\omega_{f}\cdot\alpha_{f}\big{)}.$

Here, we have added the extra factor $\textrm{Vol}(f_{*})$ to the second
summand of the right hand side of Equation (5).

##  5\. On liftings in general settings

In this section, we introduce general liftings. In Subsection 5.1 we study
liftings on regular spheres defined by integration of the differential
liftings. Further in Subsection 5.2, we discuss a slightly different notion of
liftings on the Grassmannians of planes.

For simplicity we deal with polytopal embeddings of complexes that also cover
all 1-frameworks without self-intersections. The theory for self-intersecting
frameworks is very similar, so we omit it here. The problem for general
complexes remains open.

###  5.1. Liftings on regular spheres

In this subsection, we define liftings in the class of regular $k$-spheres in
$\mathbb{R}^{n}$ (i.e. differential embeddings of the unit $k$-sphere with
maximal ranks of Jacobi matrices at every point; note that we allow self-
intersections). In this section we do not consider the case $k=0$.

####  5.1.1. Definition of liftings

Denote the space of regular $k$-spheres passing through $p$ by
$\Omega_{k}(p)$.

######  Definition 5.1.

Consider a regular sphere $\gamma$ in $\Omega_{k}(p)$. The homotopy
corresponding to the linear homothety of $\gamma$ (at time $1$) to $p$ (at
time $0$) is called a cone homotopy. We denote it by
$\mathcal{H}_{p}(\gamma)$.

######  Definition 5.2.

Let $\alpha$ be a decomposable $m$-form defined on $S\subset\mathbb{R}^{n}$
and a point $r\in S$. Consider some oriented orthonormal basis
$(e_{1},\ldots,e_{n})$ in which we have

$\alpha|_{r}=f\cdot e_{1}\wedge\ldots\wedge e_{m}.$

We say that the form

$\beta|_{r}=f\cdot e_{m+1}\wedge\ldots\wedge e_{n}$

is the Hodge star of $\alpha$ at $p$.

An arbitrary $m$-form $\alpha$ can be represented as a sum of decomposable
forms. Here we define the Hodge star for $\alpha$ by additivity.

We say that the form $\beta$ is the Hodge star of $\alpha$ if it is the Hodge
star at every point of $S$. Denote the Hodge star by $\star\alpha$.

######  Definition 5.3.

Consider an $m$-dimensional polytopal complex $C$ in $\mathbb{R}^{n}$ with a
force-load $\omega$ on the associated framework $\mathcal{F}_{C}$. Let $n>m+1$
and let $p$ be in the complement of $\mathcal{F}_{C}$. Consider the function
$L_{\mathcal{F}_{C},\omega,p}$ defined on the set of all regular spheres of
dimension $n-m-1$ as follows:

$L_{\mathcal{F}_{C},\omega,p}(\gamma)=\int\limits_{\mathcal{H}_{p}(\gamma)}\star\alpha[C,\omega].$

We say that the function $L_{\mathcal{F}_{C},\omega,p}$ is the lifting of
$(\mathcal{F}_{C},\omega)$.

######  Remark 5.4.

Liftings are continuous in the continuous topology on $\Omega_{n-m-1}(p)$.
There is no significant dependence on the choice of the point $p$ (due to
Proposition 5.5 below).

Let us collect several simple statements on integration of differential forms
over regular embeddings without boundary.

####  5.1.2. Co-dimension 0 obstacles cycle integration

Let us start with the following proposition.

######  Proposition 5.5.

Let $H_{1}$ and $H_{2}$ be two regular homotopies from $\gamma_{1}$ to
$\gamma_{2}$ for the regular $(n-m-1)$-spheres, and let $\alpha$ be any
constant $(n-m)$-form in $\mathbb{R}^{n}$. Then

$\int_{H_{1}}\alpha=\int_{H_{2}}\alpha.$

###### Proof.

Let $H\times I$ be a regular homotopy between two regular homotopies $H_{1}$
and $H_{2}$. Then by Stokes’ formula (see, e.g. in [1]) we have

$\int_{H_{1}}\alpha-\int_{H_{2}}\alpha=\int_{H\times I}d\alpha=\int_{H\times
I}0=0.$

This implies the statement of the proposition. ∎

####  5.1.3. Co-dimension 1 obstacles cycle integration

Let us continue with the following definition.

######  Definition 5.6.

Let $\ell$ be an $m$-dimensional plane in $\mathbb{R}^{n}$, we say that a
Heaviside form on the space of regular $m$-spheres with respect to $\ell$ is
defined by the following formula

$\chi_{\ell}(\gamma)={\rm lk}(\ell,\gamma)\cdot\star d{\ell},$

where $d\ell$ is the volume $m$-form on $\ell$.

Note that $\chi_{\ell}$ is an $(n{-}m)$-form.

######  Proposition 5.7.

Let $H_{1}$ and $H_{2}$ be two regular homotopies from $\gamma_{1}$ to
$\gamma_{2}$ for the regular $(n{-}m{-}1)$-spheres in $\mathbb{R}^{n}$, and
let $\ell$ be an $m$-dimensional plane in $\mathbb{R}^{n}$. Then

$\int_{H_{1}}\chi_{\ell}=\int_{H_{2}}\chi_{\ell}.$

###### Proof.

Case 1. Assume that the homotopies $H_{1}$ and $H_{2}$ from $\gamma_{1}$ to
$\gamma_{2}$ cross the plane $\ell$ at single points $p_{1}$ and $p_{2}$ (at
parameter values $t_{1}$ and $t_{2}$) respectively.

Denote the closed loop formed by $H_{1}$ and $H_{2}$ (where $H_{2}$ is passed
in the opposite direction). The parameter values $t_{1}$ and $t_{2}$ (for
$H_{1}$ and $H_{2}$) divide this loop into two parts $S_{1}$ and $S_{2}$
containing $\gamma_{1}$ and $\gamma_{2}$, respectively.

Consider the homotopy which is a parallel transform of $H_{1}(*,t_{1})$ by the
vector $p_{2}{-}p_{1}$ (here we add an infinitesimal tubular neighbourhood
from the base point for the $(n{-}m{-}1)$-spheres along the vector
$p_{2}-p_{1}$, which will not affect the value of the integrals). Denote the
corresponding path by $S_{3}$ and its inverse by $S_{4}$.

Finally, let us pick some regular homotopy that joins $H_{1}(*,t_{1})$
transported to $p_{2}$ and $H_{2}(*,t_{2})$. Denote this path by $S_{5}$ and
its inverse by $S_{6}$ (we assume that this homotopy keeps the point $p_{2}$
fixed and the other points of the homotopy do not intersect $\ell$), see
Figure 10.

$\begin{array}[]{c}\includegraphics[scale={.8}]{fig/figure-ok-2.pdf}\end{array}$ | $\qquad\longrightarrow\qquad$ | $\begin{array}[]{c}\includegraphics[scale={.8}]{fig/figure-ok-3.pdf}\end{array}$
---|---|---
Figure 10. Homotopies $H_{1}$ and $H_{2}$ and the corresponding paths
$S_{1},\ldots,S_{6}$.

Denote

$\alpha=\chi_{\ell}(\gamma_{1})\qquad\hbox{and}\qquad\beta=\chi_{\ell}(\gamma_{2})-\chi_{\ell}(\gamma_{1}).$

Then we have:

$\displaystyle\int_{H_{1}}\chi_{\ell}-\int_{H_{2}}\chi_{\ell}$
$\displaystyle=\int_{S_{1}}\alpha+\int_{S_{2}}(\alpha+\beta)$
$\displaystyle=\int_{S_{1}\cup S_{3}\cup S_{5}}\alpha+\int_{S_{2}\cup
S_{6}\cup S_{4}}(\alpha+\beta)-\int_{S_{3}}\alpha$
$\displaystyle-\int_{S_{5}}\alpha-\int_{S_{4}}(\alpha+\beta)-\int_{S_{6}}(\alpha+\beta)$
$\displaystyle=-\int_{S_{3}}\alpha-\int_{S_{4}}(\alpha+\beta)$
$\displaystyle=-\int_{S_{3}\cup
S_{4}}\alpha-\int_{S_{4}}\beta=-\int_{S_{4}}\beta=0.$

The second and the third equalities hold by Proposition 5.5. The last equality
holds since we integrate along the kernel of the form (recall that
$p_{1}p_{2}$ is orthogonal to the integral levels of the form).

Case 2. Assume that the homotopy $H_{1}$ does not cross the line $\ell$ while
$H_{2}$ crosses $\ell$ transversally at two different points $p_{1}$ and
$p_{2}$, respectively.

This case is reduced to the previous one by considering it as a single
homotopy $H_{1}$ and then $H_{2}$ based in the opposite direction. The further
proof literally repeats the previous case.

General Case is a composition of Cases 1 and 2. ∎

####  5.1.4. Co-dimension 2 obstacles cycle integration

Next we show the following monodromy statement for $(m-1)$-dimensional facets
in $m$-dimensional polytopal complexes.

######  Proposition 5.8.

Let us consider a homotopy from a point to itself by $m$-spheres along the
$(m+1)$-sphere. We assume that the convex interior of the $(m+1)$-sphere
contains the only point of facets at the center of the sphere. Then the
integral of $\star\alpha[C,\omega]$ over this sphere is zero.

###### Proof.

Let us consider the cone homotopy that shrinks the $(m+1)$-sphere to its
center along the cone. On the one hand by Propositions 5.5 and 5.7, this will
not change the value of the integral. On the other hand we are shrinking to a
single point, and hence the limiting value of the integral tends to 0.

Therefore, the integral along the $(m+1)$-sphere around a facet $e$ is zero. ∎

####  5.1.5. Continuity of liftings

Let us finally formulate the following general property of liftings.

######  Theorem 5.9.

The lifting is a continuous function in the continuous topology on
$\Omega_{n-m-1}(p)$.

###### Proof.

Let us outline the proof. Consider a small regular perturbation (within any
small distance in the space of continuous homotopies) changing the original
regular homotopy to the one avoiding all $(m-1)$-dimensional facets of the
complex and transversally intersecting the complex in finitely many points.
The resulting homotopy can be decomposed into the homotopies studied in
Propositions 5.5, 5.7, and 5.8. ∎

###  5.2. Grassmannian liftings

We conclude this section with the following remarkable representation of
liftings as functions on affine Grassmannians.

The informal idea behind the below definitions is as follows. First of all,
let us restrict the sets of regular $m$-spheres to geometric $m$-spheres
passing through a given point $p$. Secondly, let us “move” $p$ to infinity.
That will bring us to the affine Grassmannian of oriented $m$-planes ${\rm
Gr}(m,n)$.

First of all we give a general definition of distances between planes.

######  Definition 5.10.

The distance between the planes $\pi_{1}$ and $\pi_{2}$ spanned by
$(p_{1},\ldots,p_{n})$ and $(q_{1},\ldots,q_{m})$ respectively is the
following number

${\rm
dist}(\pi_{1},\pi_{2}):=\frac{\det(q_{2}-q_{1},q_{3}-q_{1},\ldots,q_{m}-q_{1},p_{1}-q_{1},\ldots,p_{n}-q_{1})}{\det(q_{2}-q_{1},q_{3}-q_{1},\ldots,q_{m}-q_{1})\cdot\det(p_{2}-p_{1},p_{3}-p_{2},\ldots,p_{n}-p_{1})}.$

Informally, distance functions serve as linear functions over Grassmannians.

In order to define the values of the liftings at all lines of the Grassmannian
we use the following notion. We consider the Grassmannian as a subset of the
projective plane $\mathbb{R}P^{N}$ defined by Plücker coordinates of planes.
So we have a natural structure of the differential manifold on Grassmannians.

######  Definition 5.11.

Let $\mathcal{F}_{C}$ be a framework of an $m$-dimensional polytopal complex
$C$ in $\mathbb{R}^{n}$. A differentiable path in ${\rm Gr}(n{-}m{-}1,n)$ is
$\mathcal{F}_{C}$-simple if it intersects $\mathcal{F}_{C}$ only finitely many
times; each of the intersections, except the very last point of the path, is
in the interior of some $m$-dimensional face and it is transversal to this
face (see Figure 11). The last line in the path can be taken arbitrarily.

Figure 11. Transversal intersections of a differentiable path with one of the
elements of $\mathcal{F}_{C}$

Let us define a lifting along $\mathcal{F}_{C}$-simple paths.

######  Definition 5.12.

Consider an $\mathcal{F}_{C}$-simple path $H$ crossing an $m$-dimensional face
$f$ of $\mathcal{F}_{C}$. We say that the sign of the crossing is the number

$\mu(H,f):={\rm sign}\Big{(}\frac{\partial({\rm dist}(H(t),f))}{\partial
t}\Big{)}.$

We say that a simple path $H$ intersects an $m$-dimensional face $f$ of
$\mathcal{F}_{C}$ with sign multiplicity $k$ if the total sum of its sign of
crossings with $f$ is $k$.

######  Definition 5.13.

Consider a framework $\mathcal{F}_{C}$ of an $m$-dimensional polytopal complex
$C$ in $\mathbb{R}^{n}$ with a force-load $\omega$. Let us fix some
$\ell_{0}\in{\rm Gr}(n{-}m{-}1,n)$ that is far away (namely it does not
intersect the convex hull of $\mathcal{F}_{C}$). Let $\ell$ be an arbitrary
plane in ${\rm Gr}(n{-}m{-}1,n)$. Let now $H$ be an $\mathcal{F}_{C}$-simple
path from $\ell_{0}$ to $\ell$. We say that the lifting at $\ell$ with respect
to the path $H$ is the following number

$L_{\mathcal{F}_{C},\omega,H}(\ell):=\sum\limits_{f\in
F_{C}}\mu(H,f)\cdot\omega_{f}\cdot{\rm dist}(\ell,f).$

We conclude this paper with the following general theorem.

######  Theorem 5.14.

The lifting $L_{\mathcal{F}_{C},\omega,H}(\ell)$ does not depend on the choice
of an $\mathcal{F}_{C}$-simple path $H$.

###### Proof.

The case of graph frameworks in $\mathbb{R}^{3}$. Here we consider the
Grassmannian ${\rm Gr}(1,3)$.

Direct calculations show that the monodromy around a trivalent vertex is
trivial. Without loss of generality we go around the three edges $p_{1}p_{2}$,
$p_{1}p_{3}$ and $p_{1}p_{4}$, where

$p_{1}=(0,0,0),\quad p_{2}=(1,0,0),\quad p_{3}=(\cos\alpha,\sin\alpha,0),\quad
p_{4}=(\cos\beta,\sin\beta,0).$

(Here $\alpha\neq\beta$ and $\alpha,\beta\in(0,\pi)\cup(\pi,2\pi)$). The
starting and the ending line is $p_{5}p_{6}$ where

$p_{5}=(a,b,0),\quad p_{6}=(c,d,1).$

In this case, the force-loads on edges $p_{1}p_{2}$, $p_{1}p_{3}$ and
$p_{1}p_{4}$ are $\lambda\sin(\alpha-\beta)$, $\lambda\sin\beta$, and
$-\lambda\sin\alpha$, respectively (for some real $\lambda$). Then the
contribution of the monodromy around the origin (passed clockwise) is as
follows:

$\begin{array}[]{l}\lambda\sin(\alpha-\beta)\cdot{\rm
dist}(p_{5}p_{6},p_{1}p_{2})+\lambda\sin\beta\cdot{\rm
dist}(p_{5}p_{6},p_{1}p_{3})-\lambda\sin\alpha\cdot{\rm
dist}(p_{5}p_{6},p_{1}p_{4}))\\\
\displaystyle\qquad=\Bigg{(}-{{\frac{b\sin\left(\alpha-\beta\right)}{\sqrt{\left(c-a\right)^{2}+\left(d-b\right)^{2}+1}}}}+{{\frac{\left(a\sin\alpha-b\cos\alpha\right)\sin\beta}{\sqrt{\left(c-a\right)^{2}+\left(d-b\right)^{2}+1}}}}-{{\frac{\sin\alpha\left(a\sin\beta-b\cos\beta\right)}{\sqrt{\left(c-a\right)^{2}+\left(d-b\right)^{2}+1}}}}\Bigg{)}\cdot\lambda\\\
\qquad=0.\end{array}$

Let us now consider a vertex of general valency $d$. Such vertex can be
decomposed locally at the vertex into the sum of trivalent vertices of the
original graph (using the techniques of quantisations introduced in [16]). Let
us illustrate it with the following example:

Any $\mathcal{F}_{C}$-simple closed path is decomposable into several
monodromies around single vertices. Hence, by additivity the value of the
corresponding lifting is zero.

Therefore, two different $\mathcal{F}_{C}$-simple paths $H_{1}$ and $H_{2}$ to
the same line $\ell$ define the same lifting (as the closed path obtained by
$H_{1}$ and then $H_{2}$ inverse provides the zero lifting). This concludes
the proof for the graph frameworks in $\mathbb{R}^{3}$.

The case of $m$-polytopal frameworks in $\mathbb{R}^{n}$. Let us sketch the
proof in this case; in fact it repeats the proof of the previous theorem.

Firstly, we consider a monodromy of parallel $(n{-}m{-}1)$-planes around some
trivalent facet $e\in E_{C}$. Here we assume that any line of $e$ is not
parallel to a line in the parallel planes. This monodromy can be projected to
the monodromy of a line around a trivalent vertex of a graph in
$\mathbb{R}^{3}$. Hence it is trivial by the above case.

Secondly, the monodromy about an arbitrary facet $e$, can be decomposed into
the sum of monodromies about trivalent copies of $e$.

Thirdly, by additivity the lifting along any $\mathcal{F}_{C}$-simple closed
path is zero.

Therefore, the liftings along different paths provide the same value. ∎

######  Remark 5.15.

One can restrict liftings to a pencil of parallel planes in Grassmannians.
Then (by identifying the points of the parallel planes) we can project the
liftings to the liftings over $\pi_{0}$ of the corresponding projected
polytopal $m$-dimensional complex in $\mathbb{R}^{m+1}$. (Here we assume that
the projection of $\mathcal{F}_{C}$ does not reduce the dimensions of its
faces.)

## Acknowledgements

The authors are grateful to C. Lescop for useful discussions. F. M. was
partially supported by the FWO grants G0F5921N (Odysseus) and G023721N, the KU
Leuven iBOF/23/064 grant, and the UiT Aurora MASCOT project. C. M. was
partially supported by the Austrian Science Fund (FWF) through grant I 4868
(SFB-Transregio “Discretization in Geometry and Dynamics”). B.S. was partially
supported by the ICMS Knowledge Exchange Catalyst Programme.

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Authors’ addresses:

Department of Mathematical Sciences, University of Liverpool, UK
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Departments of Mathematics and Computer Science, KU Leuven, Belgium
<EMAIL_ADDRESS>
Institute of Discrete Mathematics and Geometry, TU Wien, Austria
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Department of Mathematics and Statistics, Lancaster University, Lancaster, UK
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