Linear configurations of complete graphs K4 and K5 in ℝ3, and higher dimensional analogs

We investigate the space [Formula: see text] of images of linearly embedded finite simplicial complexes in [Formula: see text] isomorphic to a given complex [Formula: see text], focusing on two special cases: [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, and [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, so [Formula: see text] has codimension 2 in [Formula: see text], in both cases. The main result is that for [Formula: see text], [Formula: see text] (for either [Formula: see text]) deformation retracts to a subspace homeomorphic to the double mapping cylinder [Formula: see text] where [Formula: see text] is the alternating group and [Formula: see text] the symmetric group. The resulting fundamental group provides an example of a generalization of the braid group, which is the fundamental group of the configuration space of points in the plane.

On simple maps

This chapter deals with simple maps of finite simplicial sets, along with some of their formal properties. It begins with a discussion of simple maps of simplicial sets, presenting a proposition for the conditions that qualify a map of finite simplicial sets as a simple map. In particular, it considers a simple map as a weak homotopy equivalence. Weak homotopy equivalences have the 2-out-of-3 property, which combines the composition, right cancellation and left cancellation properties. The chapter proceeds by defining some relevant terms, such as Euclidean neighborhood retract, absolute neighborhood retract, Čech homotopy type, and degeneracy operator. It also describes normal subdivision of simplicial sets, geometric realization and subdivision, the reduced mapping cylinder, how to make simplicial sets non-singular, and the approximate lifting property.

h-COBORDISMS AND MAPPING CYLINDER OBSTRUCTIONS

In this paper we prove a realizability theorem for Quinn’s mapping cylinder obstructions for stratified spaces. We prove a continuously controlled version of the s-cobordism theorem which we further use to prove the relation between the torsion of an h-cobordism and the mapping cylinder obstructions. This states that the image of the torsion of an h-cobordism is the mapping cylinder obstruction of the lower stratum of one end of the h-cobordism in the top filtration. These results are further used to prove a theorem about the realizability of end obstructions.