Trunks and classifying spaces
- 134 Downloads
Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks . A rackX gives rise to a trunkT (X) which has a single vertex and the setX as set of edges. Therack spaceBX ofX is the realisation of the nerveNT (X) ofT(X). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural mapBX ↦BAs(X) whereBAs(X) is the classifying space of the associated group ofX. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown-Higgins classifying space of the associated crossed module [8, Section 2] and .
The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.
Mathematics Subject Classifications (1991)57M 57Q 18C 18F 55R 55U 55P
Key wordstrunk classifying space rack cubical set nerve crossed module crossed complex knot link codimension 2
Unable to display preview. Download preview PDF.
- 1.Brieskorn, E.: Automorphic sets and singularities,Contemp. Math. 78 (1988), 45–115.Google Scholar
- 2.Brown, R. and Higgins, P.: On the algebra of cubes,J. Pure Appl. Algebra 21 (1981), 233–260.Google Scholar
- 3.Brown, R. and Higgins, P.: The classifying space of a crossed complex,Math. Proc. Cambridge Philos. Soc. 110 (1991), 95–120.Google Scholar
- 4.Brown, R. and Higgins, P.: The classifying space of a crossed complex, U. Coll. of N. Wales, Math. Preprint 89.06, 1989.Google Scholar
- 5.Buoncristiano, S., Rourke, C., and Sanderson, B.: A geometric approach to homology theory,London Math. Soc. Lecture Note Ser. 18 (1976).Google Scholar
- 6.Conway, J. C. and Wraith, G. C.: Correspondence, 1959.Google Scholar
- 7.Duskin, J.: Simplicial methods and the interpretation of “triple” cohomology,Mem. Amer. Math. Soc. 3, Issue 2, (1973), 163.Google Scholar
- 8.Fenn, R. and Rourke, C.: Racks and links in codimension two,J. Knot Theory Ramifications 1 (1992), 343–406.Google Scholar
- 9.Fenn, R., Rourke, C., and Sanderson, B.: An introduction to species and the rack space, in M. E. Bozhüyük (ed.),Topics in Knot Theory, Kluwer Academic Publishers, 1993, pp. 33–55.Google Scholar
- 10.Freyd, P. and Yetter, D.: Braided compact closed categories with applications to low dimensional topology,Adv. Math. 77 (1989), 156–182.Google Scholar
- 11.Hintze, H.: Polysets, □-sets and semi-cubical sets, MPhil Thesis, Warwick, 1973.Google Scholar
- 12.Joyce, D.: A classifying invariant of Knots; the Knot quandle,J. Pure Appl. Algebra 23 (1982), 37–65.Google Scholar
- 13.James, I.: On the suspension triad,Ann. Math. 63 (1956), 191–247.Google Scholar
- 14.Kauffman, L.: Knot-crystals — classical Knot theory in modern guise, inKnots and Physics, World Scientific, 1991, pp. 186–204.Google Scholar
- 15.Kan, D. M.: Abstract homotopy I,Proc. Nat. Acad. Sci. 41 (1955), 1092–1096.Google Scholar
- 16.Matveev, S.: Distributive groupoids in Knot theory,Math. USSR-Sb. 47 (1984), 73–83.Google Scholar