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# Parity structure on associahedra and other polytopes

thesis

posted on 2022-03-28, 21:26 authored by Christopher NguyenA parity structure is a name given to a formalism of pasting diagrams, among these are: parity complexes in the sense of Street, pasting schemes in the sense of Johnson, and directed complexes in the sense of Steiner. The idea behind these formalisms is to take a set of faces and attach an orientation to each face so that we obtain a presentation of a (strict) free ω-category.
The above formalisms include examples such as the simplexes and hypercubes, but are not sufficiently general to allow for other reasonable examples. In this thesis, our main goal is to construct a parity structure on other polytopes. An example of interest is the polytope family known as the associahedra. Notable work on the associahedra includes that of Tamari and Stasheff.
One approach to associahedra is via left bracketing functions (lbf) due to Huang-Tamari. We will introduce a generalisation of an lbf which we call a higher left bracketing function (hlbf), and show that they correspond to the faces of the associahedron. We are able to construct a parity structure on the hlbfs. This parity structure does not satisfy the parity complex axioms due to Street. However, it does satisfy a modified set of axioms given by Campbell. It follows from the results of Campbell that we have a loop free pasting scheme in the sense of Johnson.
The construction of a parity structure on the associahedron is generalised to a more basic structure known as an abstract pre-polytope. To achieve this, we introduce the idea of a label structure on an abstract pre-polytope. From a label structure, we obtain a parity structure which is then proven to satisfy the axioms due to Campbell. Finally, we use this construction on other polytopes such as the hypercubes and permutohedra.