On replacing proper Dehn maps with proper embeddings
| Autor: | C. D. Feustel |
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| Rok vydání: | 1972 |
| Předmět: | |
| Zdroj: | Transactions of the American Mathematical Society. 166:261-267 |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/s0002-9947-1972-0293644-9 |
| Popis: | In this paper we develop algebraic and geometric conditions which imply that a given proper Dehn map can be replaced by an embedding. The embedding, whose existence is implied by our theorem, retains most of the algebraic and geometric properties required in the original proper Dehn map. I. Preliminaries. In this paper all spaces will be simplicial complexes and all maps will be piecewise linear. The boundary, closure, and interior of a subset X of a space Y will be denoted by bd (X), cl (X) and int (X), respectively. A proper map f, taking a space X into a space Y, is a map such that f bd (X) = bd (Y) n f(X). A Dehn map f, taking a compact, bounded, two-manifold F into a 3-manifold M, is a map such that (1) bd (F)=f-lfbd (F); (2) fIbd (F) is a homeomorphism. All unlabeled homomorphisms are natural maps induced by inclusion. We shall consider the problem of replacing a proper Dehn map of a compact, connected, orientable, bounded surface F into a compact, (necessarily) bounded, irreducible, orientable 3-manifold M with an embedding. As will be pointed out later, the requirement of orientability may not be strictly necessary. The author would like to thank J. Gross for a number of helpful conversations. In [3] Papakyriakopoulos proved Dehn's lemma, that is, a proper Dehn map f of a disk 9 into a 3-manifold M can be replaced by a proper embedding g of 9 into M such that g bd (9) =f bd (9). A natural question to consider is the following: Let f: F -* M be a proper Dehn map of a compact, connected, bounded, orientable surface F into a compact 3-manifold M. Does there always exist a proper embedding g: F1 -? M such that (1) g(bd (F1))'f(bd (F)); (2) genus (F1) |
| Databáze: | OpenAIRE |
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