F. Waldhausen defines a k-fold end structure on a space X to be an ordered k-tuple of continuous maps xj :X(--->)R('+), 1 (LESSTHEQ) j (LESSTHEQ) k (where R('+) is the euclidean half line) yielding a map x:X(--->)(R)('k). The pairs (X,x) are made into the category E('k) of spaces with k-fold end structure. Attachments and expansions in E('k) are defined by induction on k, where elementary attachments and expansions in E('0) have their usual meaning. For Z (epsilon) E('k), the category E('k)/Z consists of pairs (X,i) where i:Z(--->)X is an inclusion in E('k) such that there exists an attachment from i(z) to X. And E('k)//Z is the category whose objects are triples (X,i,r) with (X,i) (epsilon) E('k)/z and r:X(--->)Z a retraction. An infinite complex over Z is a sequence of inclusions in E('k)//Z, X = {X(,1))(Y,y) in E('k) can be madebounded with respect to equivalent k-fold end structures x',y' onX,Y respectively. When X (epsilon) S(,1)(R('k)), that fact can be used to extendthe guaranteed deformation X(SQUIGARR)R('k) in E('k) to a proper deformation(')X(SQUIGARR)D('k) where / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / is the associated compactification of X. It is shown that after embedding (')X in R('n) for n large enough, and choosing a regular neighborhood (')N of (')X, that ((')N,D('k)) is a proper unknotted ball pair. The result proves, when R('k) is given the natural product k-fold end structure, Waldhausen's group S(,1)(R('k)) = 0. An exact sequence established by M. Petty is applied to show S(,0)(R('k)) is also trivial. As a consequence, we show that when X is a generalized q-manifold (q (GREATERTHEQ) 5) with singular set S(X) a polyhedron, XxR a piecewise-linear (q+1)-manifold, then X is the cell-like image of a manifold. / Source: Dissertation Abstracts International, Volume: 43-12, Section: B, page: 4012. / Thesis (Ph.D.)--The Florida State University, 1982.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_75011 |
| Contributors | KUTTER, MARY YEILDING., Florida State University |
| Source Sets | Florida State University |
| Detected Language | English |
| Type | Text |
| Format | 66 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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