/pdf/on-irreducible-3-manifolds-which-are-sufficiently-large-1dwm1gl243.pdf

On irreducible 3-manifolds which are sufficiently large

The following question has been posed by Bing [1]: "Which compact, connected 3-manifolds can be obtained from S3 as follows: Remove a finite collection of mutually exclusive (but perhaps knotted and linking) polyhedral tori T1, T2, * *, To from S3, and sew them back. " This paper answers that question by showing that every closed, connected, orientable, 3-manifold is obtainable from S3 in the above way. Whereas this fact can now be deduced from general theorems of differential topology, the combinatorial proof given here is direct and elementary; while, in the proof, a study is made of a certain type of homeomorphism of a two dimensional manifold that is of interest in itself. Having obtained the above mentioned result on 3-manifolds, it is then easy to deduce the well known result (Theorem 3) that the combinatorial cobordism group for orientable 3-manifolds is trivial.

http://www.jstor.org/stable/pdfplus/1970373.pdf

A Representation of Orientable Combinatorial 3-Manifolds

Objekt dieser Arbeit sind orientierbare kompakte 3-dimensionale Mannigfaltigkeiten, M, mit der Eigenschaft: Es gibt in A~/ein System disjunkter Tori, T, so dab jede Komponente yon M U(T) hom6omorph ist zu einem Faserbfindel mit S 1 als Faser (U(T) bezeichnet eine regul/ire Umgebung von T). Wir bezeichnen T als eine Graphenstruktur f/Jr M, und M selbst als eine Graphenmannigfaltigkeit (als AnlaB dieser Namensg ebunK kann man .~9 betrachten). Es ist f/Jr unsere Zwecke notwendig, eine vorgegebene Graphenstruktur zun/ichst in bestimmter Weise zu vereinfachen (dabei wird i.a. die Mannigfaltigkeit in eine zusammenh/ingende Summe zerlegt). Eine Graphenstruktur, die sich nicht weiter vereinfachen 1/iBt, heil3t reduzierte Graphenstruktur, und eine Mannigfaltigkeit mit vorgegebener reduzierter Graphenstruktur heiBt reduzierte Graphenmannigfaltigkeit. Unser Hauptergebnis ist: Ein Hom6omorphismus zwischen ,,hinreichend komplizierten" reduzierten Graphenmannigfaltigkeiten 1/igt sich so deformieren, dab er die eine Graphenstruktur auf die andere abbildet; (8.1). Spezielle Graphenmannigfaltigkeiten sind die orientierbaren unter den yon SEIFERT in [12] behandelten ,,gefaserten R/iumen" (Kenntnis der Seifertschen Arbeit [12] ist ffir das Verst/indnis dieser Arbeit nicht notwendig, aber nfitzlich); unser Ergebnis impliziert, dab jeder Hom6omorphismus zwischen ,,hinreichend komplizierten" orientierbaren gefaserten R/iumen in einen fasertreuen Hom6omorphismus deformiert werden kann; (10.1). Die von SEIFERT angegebene Klassifikation der gefaserten R~iume gegenfiber fasertreuen Hom6omorphismen 1/iBt sich fibertragen auf die Klassifikation yon reduzierten Graphenmannigfaltigkeiten gegeniiber solchen Hom6omorphismen, die die Graphenstruktur respektieren (w 9); nach dem genannten Ergebnis ist dies ,,i. a." zugleich die topologische Klassifikation. Die Herleitung des Hauptsatzes beansprucht den gr6Bten Teil der Arbeit; man k6nnte sie bezeichnen als eine Anwendung der Theorie ,,inkompressibler Fl~ichen": Sei F ein System yon (orientierbaren kompakten) Fl~ichen in der (orientierbaren kompakten) Mannigfaltigkeit M (so dab FnOM=OF);

/pdf/eine-klasse-von-3-dimensionalen-mannigfaltigkeiten-i-22kdn91i2u.pdf

Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I

Consider an oriented 2-dimensional manifold m imbedded as a subcomplex in a triangulated oriented 4-dimensional manifold M in such a way that the boundary of m is contained in the boundary of M and the interior of m is contained in the interior of M. We will assume that M is a \"piecewise linear manifold\": that is, the star neighborhood of any point should be piecewise linearly homeomorphic to a 4-simplex. One can measure the local singularity of the imbedding at an interior point x of m as follows. Let N denote the star neighborhood of x in M. The boundary S=dN of N is a 3-sρhere with an orientation inherited from that of M, and k=mΓϊdNis a 1-sphere with an orientation inherited from that of m. The oriented knot type K of the imbedding of k in S is called the singularity of the imbedding at x. When k is of trivial type in dN we may say that the singularity is 0 or that x is a non-singular point or that m is locally flat at x. A surface m is called locally flat if it is locally flat at each of its points. REMARK. The singularity of m at x is clearly a combinatorial invariant of M,m,x\\ that is it is not altered if we subdivide M rectilinearly. We do not know whether or not this singularity is a topological invariant, except in the special case of a locally flat point. The topological invariance of the concept of local flatness is easily proved, making use of Dehn's lemma, [12, §28(i)]. Of course the local singularity can also be measured at a boundary point x. In this case 3iVis a 3-cell, mΓϊdN is a 1-cell spanning it, and the singularity is a type of spanning 1-cell. In this paper we shall consider only imbeddings whose boundary points are all non-singular. Since a singular point must be a vertex in any triangulation of the pair mczM the singular points are always isolated. If m is compact (as it will be from now on) there can therefore be only a finite number of singular points. For the rest of this paper m will be a 2-sphere and M will be the 4-dimensional euclidean space R\\ that is, the 4-sphere punctured at oo. The basic problem

/pdf/singularities-of-2-spheres-in-4-space-and-cobordism-of-knots-mitq65h9zw.pdf

Singularities of 2-spheres in 4-space and cobordism of knots

It is known that, for n ≤3, ‘ought to have’ can truthfully be replaced by ‘has’ (see (4), and (5), Cor. 6·6). In this paper, this conjecture will be proved for n ≥ 5. The only unsolved case then will be in dimension four.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/36726322C2D6AB5501941A68B41B74AF/S0305004100036756a.pdf/the-piecewise-linear-structure-of-euclidean-space.pdf

V. K. A. M. Gugenheim

Papers

Piecewise Linear Isotopy and Embedding of Elements and Spheres (II)