  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.

Metric Geometry;Combinatorics