We show that a tame ended stratified space X is the interior of a compact stratified space if and only if a K-theoretic obstruction γ * (X) vanishes. The obstruction γ * (X) is a localization of Quinn's mapping cylinder neighborhood obstruction. The main results are Theorem 1.6 and Theorem 1.7 below. In particular, this explains when a G-manifold is the interior of a compact G-manifold with boundary. Our methods include a new transversality theorem, Corollary 1.17.
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.
We present a concise, yet self-contained module for teaching the notion of area and the Fundamental Theorem of Calculus for different groups of students. This module contains two different levels of rigour, depending on the class it used for. It also incorporates a technological component.
Can sixth graders learn trigonometry? Although this material is taught at the high school and college levels, most sixth graders are smart enough to learn it. In fact, South Korean, Romanian, and American schools abroad introduce basic trigonometry in middle school. U.S. schools can do it, too. With the right preparation, any student can learn a bit of trigonometry; a simple curiosity is enough to get started.
A stratified space is a filtered space with manifolds as its strata. Connolly and Vajiac proved an end theorem for stratified spaces, generalizing earlier results of Siebenmann and Quinn. Their main result states that there is a single K-theoretical obstruction to completing a tame-ended stratified space. A necessary condition to completeness is to find an exhaustion of the stratified space, i.e. an increasing sequence of stratified spaces with bicollared boundaries, whose union is the original space. In this paper we give an example of a stratified space that is not exhaustible. We also prove that the Connolly-Vajiac end obstructions can be realized.
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