Triple chords and strong (1, 2) homotopy

Strong Homotopy Types, Nerves and Collapses

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relati...

1 1 Ju l 2 00 0 UNIVERSAL HOMOTOPY THEORIES

Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the Dwyer-Kan theory of framings, to sheaf theory, and to the homotopy theory of schemes.

Strong Homotopy Equivalence of 3-manifolds

On Strong Homotopy for Quasi-schemoids

A quasi-schemoid is a small category with a particular partition of the set of morphisms. We define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. The homotopy set of self-homotopy equivalences on a quasi-schemoid is used as a homotopy invariant in the study. The main theorem enables us to deduce that the homotopy invariant for the quasi-schemoid in...

Homotopy invariants and time evolution in (2+1)-dimensional gravity.

We establish the relation between the ISO(2, 1) homotopy invariants, and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2, 1) cycle conditions, are equivalent to the ISO(2, 1) cycle conditions for the representations ρ : π 1 (Σ g,N) → ISO(2, 1). Also, the symplectic structure on the space of invariants is closely related to that of ...