2023/12/15 by Oleg Karpenkov, Fatemeh Mohammadi, Karpenkov, Oleg +4 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #05C10 #52C25 #57Q99 #Cellular Mechanics and Interactions #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2312.09891
openalex publication_date 2023/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≥ 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,…, d). Finally we discuss a natural representation of generalised liftings as real-valued functions on Grassmannians.