h-COBORDISMS AND MAPPING CYLINDER OBSTRUCTIONS

Abstract

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.2000 Mathematics subject classification: primary 57Q10; secondary 57Q20, 57Q40, 57N40, 57N80.

Realization theorems for end obstructions

“…We follow an approach similar to the ones of Quinn ( [29,30]) and Ranicki ([34, 35]). The full details are provided in [40] and [41].…”

“…Here the categorical constructions Ad Z[Π(X)] mean the abelian category generated by Π(X). See [41] for a detailed description.…”

“…We will recall their definitions, using the algebraic approach of [33] and [35]. See also [40] and [41] for more details. (If B is compact, then and δ can be chosen to be positive real numbers; in this case the K 0 groups will not have the locally finite superscript.…”

“…The structure of this paper is the following: in Section 2 we review the theory of geometric modules and continuously controlled K-theory. More details are provided in [41]. Section 3 recalls the definition of the finiteness obstruction as introduced by F. Quinn in [30].…”

“…We provide a quick account, for the reader's convenience. More details are provided in [41]. We also discuss the relation between the torsion of an h-cobordism and the mapping cylinder obstructions.…”