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**Table of contents**

References [1] M. Ahmad and J. Theory and application, Theory and Applications of Math. Baikov, R.

Gilmanov, I. Taimanov, and A. Halzinger et. Baldomir and P. Barata, P. Santos N. Bernardo, and A. Birman and W. Menasco, Studying links via closed braids. V: the unlink, Trans. Di Concilio, C. Guadagni, J. Peters, and S. Ramanna, Descriptive proximities I: Properties and interplay between classical proximities and overlap, arXiv , no. Cooke and R. Finney, Homology of cell complexes.

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## Cellular homology

Cui, Video vortex cat cycles part 1, Tech. Dareau, E. Levy, M. Aguilera, R. Bouganne, E. Akkermans, F. Gerbier, and J. Beugnon, Revealing the topology of quasicrystals with a diffraction experiment, arXiv, Physical Review Letters , no. Dzedolik, Vortex properties of a photon flux in a dielectic waveguide, Technical Physics 75 , no. Dvali E. Adelberger and A. Gruzinov, Structured light meets structured matter, Phys. Letters 98 , —1——4. Edelsbrunner and J. Harer, Computational topology. An introduction, Amer. Fermi, Persistent topology for natural data analysis - a survey, arXiv , no.

Flammini and A. Stasiak, Natural classification of knots, Proc. A Math. Guadagni, Bornological convergences on local proximity spaces and wm -metric spaces, Ph. Di Concilio, 79pp. Ostavari H. Boomari and A. Zarei, Recognizing visibility graphs of polygons with holes and internal-external visibility graphs of polygons, arXiv , no. Hance, Algebraic structures on nearness approximation spaces, Ph. Thomson Lord Kelvin , On vortex atoms, Proc. Leader, On clusters in proximity spaces, Fundamenta Mathematicae 47 , — Litchinitser, Structured light meets structured matter, Science, New Series , no.

Maehara, The Jordan curve theorem via the Brouwer fixed point theorem, Amer. Monthly 91 , no. Munkres, Topology, 2nd ed. Murphy and D. MacManus, Ground vortex aerodynamics under crosswind conditions, Experiments in Fluids 50 , no. Pellikka, S. Suuriniemi, and L. Kettunen, Homology in electromagnetic boundary value problems, Boundary Value Problems , no. The concept of cochains allows the association of numbers not only to single cells, as chains do, but also to assemblies of cells.

Briefly, the necessary requirements are that this mapping is not only orientation-dependent, but also linear with respect to the assembly of cells, modeled by chains.

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A cochain representation is now the global quantity association with subdomains of a cell complex, which can be arbitrarily built to discretize a domain. Physical fields therefore manifest on a linear assembly of cells. Based on cochains, topological laws can be given a discrete representation. The space of all linear mappings on is denoted by , where the elements of are called cochains. Cochains express a representation for fields over a discretized domain. Addition and multiplication by a scalar are defined for the field functions and so for cochains.

To extend the expression possibilities, coboundaries of cochains are introduced. Then, the following sequence with is generated:. Figure: Cochain complex with the corresponding coboundary operator: 1. The algebraic structure of chains is an important concept, e.

## Homology of Cell Complexes

A chain which is an element of is a k-cycle. A chain which is an element of is a p-boundary. As can be seen, the boundary operator expression yields Figure 1. The first homology group [ 53 ] is the set of closed 1-chains curves in a space, modulo the closed 1-chains which are also boundaries. In the remainder of this work the ring will be or , in which case the modules are vector spaces or Abelian groups, respectively, e. To give an example, the first homology group is the set of closed -chains curves modulo the closed -chains which are also boundaries. This group is denoted , where are cycles or closed 1-chains and are -boundaries.

Another example is given in Figure 1. Geometrically, however, and are distinct [ 25 ] despite an isomorphism. An element of is a formal sum of -cells, where an element of is a linear function that maps elements of into a field. Chains are dimensionless multiplicities, whereas those associated with cochains are physical quantities [ 35 ].

The extension of cochains from single cell weights to quantities associated with assemblies of cells is not trivial and makes cochains very different from chains, even on finite cell complexes.

### References

Nevertheless, there is an important duality between -chains and -cochains. For a chain and a cochain , the integral of over is denoted by , and integration can be regarded as a mapping, where represents the corresponding dimension:. Integration in the context of cochains is a linear operation: given and , reads.

Reversing the orientation of a chain means that integrals over that chain acquire the opposite sign. The boundary homomorphism is: 1.