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Representation of manifolds. (English) JFM 54.0611.01

Der Brouwersche Abbildungsgrad, der ursprünglich für stetige Abbildungen von simplizialen Mannigfaltigkeiten definiert wurde, wird hier allgemeiner für abgeschlossene “topologische” Mannigfaltigkeiten (Dieser Ausdruck rührt von H. Hopf [vgl. das folgende Referat] her. Verf. verwendet hierfür die Bezeichnung: “lokal simpliziale Mannigfaltigkeit”) erklärt, d. h. für kompakte zusammenhängende topologische Räume, die ein System von Umgebungen besitzen, deren jede mit dem Innern einer Euklidischen \(n\)-dimensionalen Kugel homöomorph ist. Die Frage, ob sich derartige Räume “triangulieren lassen”, also selbst simpliziale Mannigfaltigkeiten sind, ist bekanntlich offen. Die wesentlichen Eigenschaften des Abbildungsgrades – in erster Linie die Multiplikation der Abbildungsgrade bei Zusammensetzung von mehreren Abbildungen – bleiben bei dieser Verallgemeinerung erhalten.
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[1] Brouwer, ??ber Abbildung von Mannigfaltigkeiten?, Math. Annalen 71, p. 97; this work is referred to in the sequel as ?A. v. M.?.
[2] ?A. v. M.?, p. 97. By ann-dimensional element is understood the topological image of ann-dimensional simplex and by a simplex star ofR n a finite number of simplexes dense in a neighbourhood of a common vertexO, no two of which have inner points in common and any two of which have a commonp-dimensional face (0?p?n?1) but no further common point.
[3] ?Beweis der Invarianz desn-dimensionalen Gebiets?, Math. Annalen71, p. 305 and 306; the definition there given is here completed in accordance with a verbal communication from Prof. Brouwer.
[4] ?A. v. M.?, p. 97-98.
[5] ?A. v. M.?, p. 100.
[6] ?A. v. M.?, p. 101.
[7] Hausdorff, ?Grundz?ge der Mengenlehre? (1914), p. 213; this work is referred to in the sequel by ?Hausdorff?.
[8] Hausdorff, ?Grundz?ge der Mengenlehre? (1914), p. 260.
[9] Such manifolds have been defined by Weyl, ?Die Idee der Riemannschen Fl?che? (1913), p. 17-18; Tibor Rad?, ??ber den Begriff der Riemannschen Fl?che?, Acta litt. scient. Reg. Univ. Franc. Jos., 2, Fasc. II (1925), who proves that a closed, locally simplicial, 2-dimensional manifold is simplicial; and v. Ker?kj?rt?, ?Vorlesungen ?ber Topologie I? (1923), Einleitung, p. 5-6.
[10] These remarks have been suggested to me by Prof. Brouwer.
[11] ?Zum Metrisationsproblem?, Math. Annalen94, p. 310, Hauptsatz; extended by Tychonoff to regular spaces, Math. Annalen 95.
[12] ?A. v. M.?, p. 106, Satz 1; in the above terminology this theorem may be written: If a closed, two-sided,n-dimensional, simplicial manifold ? be uniquely and continuously represented on a simplicialn-dimensional manifold ??, there exists a finite whole numberc, invariant under continuous modification of the representation, with the property that the image of ? covers every region of ?? altogetherc times positively; if ?? be one-sided or openc is always zero. This numberc is called the degree of the representation.
[13] As with simplicial manifolds, Brouwer, ?A. v. M.?, p. 100, assign to each element of a locally simplicial manifold, a regular Euclidean simplex of fixed length of edge as its ?representative simplex?, and let there be a topological correspondence between the element and its representative simplex; then by a segment, segment path, component simplex, (n-1)-dimensional simplex inE?, is understood the image of a segment, segment path, component simplex, (n-1)-dimensional simplex respectively, in the representative simplex ofE?.
[14] Brouwer, ?Beweis desn-dimensionalen Jordanschen Satzes?, Math. Annalen71, p. 317, footnote.
[15] Brouwer ?Beweis desn-dimensionalen Jordanschen Satzes?, Math. Annalen71, p. 314.
[16] By an (n-1)-dimensional sphere ofE? is understood the image inE? of an (n-1)-dimensional sphere in the representative simplex ofE?.
[17] ?A. v. M.?, p. 101.
[18] ?A. v. M.?, p. 100.
[19] ?A. v. M.?, p. 108. The indicatrix of the (n-1)-dimensional simplexA 1 A 2 ...A n considered as a face of the simplexA 1 A 2...A n+1 is defined to beA 1,A 2, ...,A n whereA 1,A 2, ...,A n ,A n+1 is the indicatrix ofA 1 A 2...A n+1
[20] Brouwer, ??ber Jordansche Mannigfaltigkeiten?, Math. Annalen71, p. 323, ? 4.
[21] Brouwer, ??ber Jordansche Mannigfaltigkeiten?, Math. Annalen71, p. 323, ? 4, Satz 4; we have above used a particular case of this theorem.
[22] Such an element exists whenE? andE? have inner points in common Hausdorff, Axiom (B).
[23] We are here using Brouwer’s generalized indicatrix, ??ber Jordansche Mannigfaltigkeiten?, p. 324, ? 5; for the extension of the indicatrix conception to locally simplicial manifolds 1 am indebted to Prof. Brouwer personally.
[24] See remark 2 above.
[25] ?A. v. M?, p. 101-105.
[26] W. Wilson, ?Representation of a simplicial manifold on a locally simplicial manifold?, Amsterdam Proceedings29 (1926), p. 1129 sqq.; for the leading idea of the proof there given the writer was indebted to a remark on Prof. Brouwer.
[27] ?A. v. M.?, p. 106.
[28] ?A. v. M.?, p. 106, Satz 1; in the above terminology this theorem may be written: If a closed, two-sided,n-dimensional, simplicial manifold ? be uniquely and continuously represented on a simplicialn-dimensional manifold ??, there exists a finite whole numberc, invariant under continuous modification of the representation, with the property that the image of ? covers every regioni of ?? altogetherc times positively; if ?? be one-sided or openc is always zero see footnote13) above.
[29] Tibor Rad?, loc. cit. ??ber den Begriff der Riemannschen Fl?che?, Acta litt. scient. Reg. Univ. Franc. Jos., 2, Fasc. II (1925), Hilfsatz 1; see also remark 2 in the introduction.
[30] That is,V i (2) is a simplex ofE 2 with the same vertices asV i ; we recall (footnote14) as with simplicial manifolds, Brouwer, ?A. v. M.?, p. 100, assign to each element of a locally simplicial manifolds, a regular Euclidean simplex of fixed length of edge as its ?representative simplex?, and let there be a topological correspondence between the element and its representative simplex; then by a segment, segment path, component simplex, (n-1)-dimensional simplex inE?, is understood the image of a segments, segment path, component simplex, (n-1)-dimensional simplex respectively, in the representative simplex ofE?. above) that a simplex ofE 2 is the image inE 2 of a simplex in the representative simplex ofE 2.
[31] By the boundary of a set of simplexes among which the incidence relations are assigned is understood the set of those (n-1)-dimensional faces which are incident with only one simplex.
[32] Use is being here made of the ?gemischte Zerlegung? of Brouwer, ?Erweiterung des Definitionsbereichs einer stetigen Funktion?, Math. Annalen79, p. 210.
[33] This multiplication of elements was suggested by the duplication of elements used by Brouwer. ?Transformations of Projective Spaces?, Amsterdam Proceedings29 (1926), No. 6.
[34] Since the values ofp j ?q j on different sides ofF differ by unity, and on that side ofF not inX j both numbersp andq are zero.
[35] Such a chain of simplexes shall be referred to briefly as a chain of simplexes or merely as a chain.
[36] By {U i (k) } is understood the set of all simplexesU i (k) ,i. e., the suffixi takes all values for which the vertices ofU i are all inE k ;e. g. in {U i (1) } the summation is extended over a different set of the suffixesi from that in {U i (2) }; similarly with {V i (k) }.
[37] This is the ?regular subdivision? of Veblen, Analysis Situs, p. 85-86.
[38] In the sense of Hausdorff, ?Grundz?ge der Mengenlehre?, p. 260-261
[39] By {U ij (1) } is understood the set of all component simplexes of {U i (1)}, and, as in previous paragraphs, by {U i (1) } the set of all existing simplexesU i (1) ; similarly for {V ij (1) }.
[40] As in ? 9, by {1 U ij (2) } is understood the set of all component simplexes of {1 U i (2) }; similarly for {1 V ij (2) }.
[41] ??ber Jordansche Mannigfaltigkeiten?, Math. Annalen71, p. 320, ?? 5 and 6.
[42] Brouwer, ??ber Jordansche Mannigfaltigkeiten?, p. 324 and the remarks on p. 598.
[43] Brouwer, op. cit. ??ber Jordansche Mannigfaltigkeiten?, p. 324, footnote.
[44] Brouwer, op. cit. ??ber Jordansche Mannigfaltigkeiten?, Satz 6.
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