A general theory of polyhedral sets and the corresponding T-complexes

David W. Jones

GDML_Books, (1988), p.

Résumé

PrefaceThis paper is essentially David Jones' 1984 University of Wales Ph. D. Thesis, "Poly-T-complexes". It is published concurrently with Asley, 1988.The main aim is to find a setting for the most general kinds of geometrically defined compositions. Thus it comes under the slogan: "Find an algebraic inverse to subdivision". In the background is the Generalised Van Kampen Theorem, whose proof uses in an essential way general compositions of cubes. An even older background is the idea in topology of cycles in a space as some kind of composition of small pieces.The technicalities of even this stage of the theory mean that a number of problems are left unresolved. These are summarised in David Jones' final chapter. However the fundamental nature of the basic ideas should make this paper a useful stimulus to further work.N. Ashley, 1988, Simplicial T-complexes: a non-abelian version of a theorem of Dold and Kan, Diss. Math. 265.Ronald Brown, Bangor, 1988

CONTENTSPreface......................................................................................................5Introduction...............................................................................................61. A class of model categories.................................................................101.1 Categories with models......................................................................101.2. Cone-complexes...............................................................................121.3. Three standard constructions..........................................................141.4. Marked cone-complexes and the category Poly...............................161.5. Consequences of the marked face structure of a polycell................191.6. Subcategories of Poly appropriate as model categories..................232. Posets and shellability.........................................................................262.1. Shelling............................................................................................262.2. The class ET of model categories....................................................292.3. Face posets of S-polycells...............................................................322.4. The equivalence SPoly → SPos.......................................................363. Equivalences of categories of T-complexes.........................................403.1. T-complexes.....................................................................................403.2. The isomorphism ∆TC → ∆₁TC........................................................423.3. Structures and collapsing.................................................................453.4. Particular collapses in Sd X and VX..................................................473.5. The collapse $A(∆^n)$ in Sd $∆^n - p ∆^n$.......................................553.6. The functor $e_ℳ$, from ∆₁T-complexes to ℳ T-complexes...........623.7. The natural equivalence $r_ℳ ∘ e_ℳ ≃1$.....................................633.8. The equivalence of categories.........................................................654. Degeneracy structures in ℳ T-complexes...........................................674.1. An approach to degeneracy structures in ℳ T-complexes...............684.2. Pseudocylinder structures on SC-complexes...................................704.3. Rectifiers on pseudocylinders..........................................................744.4. Degenerate elements in an ℳ T-complex........................................774.5. Functors between categories of T-complexes..................................834.6. Suggested proof of Claims 5.4 and 5.10..........................................865. Comments and possibilities for further work........................................92Appendix. S-shellability of cone-complexes.............................................98Glossary of symbols..............................................................................106References...........................................................................................109

Errata Page, line: 5⁴ For: Asley, 1988 Read: Ashley, 1988

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Zbl 1002.57504

EUDML-ID : urn:eudml:doc:268464