discrete vector fields (R. Forman) and markings on cell complexes (D.W. Jones)

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

Graham Ellis has drawn my attention to the notion of *discrete vector
field* on cell complexes due to R. Forman, and on which a number of
papers may be found. Graham has some notes for the lecture he gave at
Bedlewo, June 24-30, "Dynamics, Topology, Computation".

   http://hamilton.nuigalway.ie/Berlin/bedlewo.pdf

which uses this notion.

I found, and Graham agreed,  that this notion is equivalent to the notion of "marking" for a cell complex developed in David W Jones Bangor PhD Thesis, (1984), published as

"A general theory of polyhedral sets and the corresponding {$T$}-complexes}.{Dissertationes Math. (Rozprawy Mat.)} \textbf{266} (1988) 110pp.*MR0968920*

The initial problem solved in this thesis was to define an appropriate notion of "polycell" which would allow a notion of "poly-set" as a contravariant functor from a category of poly cells and inclusions of faces, these polycells to be sufficiently general to include,  say, rhombic dodecahedra, and also the cells which occur in van Kampen diagrams for groups. (One thinks also of Stasheff's polytopes.)

Such cells had to be regular cell complexes, with one top dimensional cell, and to rigidify the notion such a cell, and all it's subcells,  were to be given the structure of a cone on its boundary.

Then, and this is the key notion, each cell was to have a marked (distinguished) face. This is equivalent to an arrow pointing from the interior of the cell to that marked face (the opposite direction to Forman's notion!). A final condition required for the further development of the theory was that of shellability. There is more discussion of this work on http://ncatlab.org/nlab/show/T-complex

Thus the marking notion is seen to be equivalent to Forman's notion. I feel the reaction should be that if a notion occurs independently and from quite different viewpoints, then the notion is not just twice as good but maybe four or more times as good! (I got this kind of gut feeling in 1967 when George Mackey told me of his work on groupoids in ergodic theory, after a lecture mine on the fundamental groupoid and van Kampen's theorem.)

I hope people will be able to look at David Jones' thesis, and other related work he developed there,  and see if the work is  useful in the increasing applications of discrete vector fields. I hope  also the motivation behind his thesis can be developed further.

Are these methods related to those of opetopes in higher category theory?

Ronnie Brown

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