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References
Dehn and Heegaard, Encykl. der Math. Wiss. III, AB3Analysis Situs; H. Kneser, Proc. Amsterdam27 (1924), p. 601; E. Bilz, Math. Zeitschr.18 (1923), S. 1.
The Foundations of Combinatory Analysis Situs, I and II, Proc. Amsterdam29 (1926), p. 611 and 627, referred to asFI andFII. See also “Additions and Corrections” to these papers, Proc. Amsterdam30 (1927), p. 670, cited asF. Add.
The definitions in § 1 are taken, with certain slight modifications, fromFI andFII, where proofs of the theorems will be found It should be noticed that Theorem 10 ofFII is not assumed. The present paper replaces § 9 ofFII.
P. Alexandroff pointed out almost simultaneously (Math. Annalen96 (1926), p. 489) that ann-simplex may be regarded as being simply itsn+1 vertices. See also the same author's paper, Math. Annalen94 (1925), p. 296.
The corresponding notation for sets of points was introduced by Carathéodory,Reelle Funktionen, p., 28.
The array Γ1 contains the array Γ2 if every unit of Γ2 is a unit or component of Γ1.
d (Γ) means “the dimension-number of Γ”.
The letters Γ and Δ will be used for arrays of uncertain character:S, T, U, V, for simplexes; small Greek letters for vertices. If a lower index is present, it denotes the dimension number; upper indices are merely distinguishing marks.
This differs from the definition ofFI in the case of unbounded arrays, but the two definitions are equivalent in the case ofmanifolds in view ofFI 21 andFII Theorem 5 “Γ→Δ” is no longer to be read “Γ is topologically equivalent to Δ” but,e. g., as “Γ leads to Δ”. Cf.F. Add., under “Topological equivalence”.
Theorems 1, 3, 5, give all the general relations between the three moves. The only other plausible suggestion — “If,M 1→M 2 and\(\bar M^1 \) is\(\bar M^2 \) then\(M^1 \xrightarrow[3]{}M^2 \)” —is false. SeeF. Add. p. 671.
The conventions governing the use of letters may be extended to assemblies;e. g. the use of M to denote ann-assembly implies that the sum of itsn-pieces is a manifold.
The invariance of the property, of being a manifold was proved by Weyl (Revista di Matem. Hisp.-Amer., 1923) by a process somewhat similar to that here adopted. Weyl's fundamental definitions are, however, of a radically different character.
Quoted on p. 402.
Two arrays arecongruent if they are completely similar.
If Γ is any assembly every component of Γ is a unit or internal component of some piece.
The connection between the Δ-and O-entities having once been cleared up it should be possible to drop the prefixes in most contexts.
A sequence of cellsa i ,a i +1, ...,a j , in which each cell bounds its successor is called a sequenceascending from a i ordescending froma j .
Themargin of ann-set of cells may be defined analogously to the margin of an array (p. 400) and has properties similar to those of the boundary.
“The boundary of ○{a k }” may be abbreviated to “the boundary ofa k ” and denoted bya k .
It will be noticed that the only property of ○ spheres used in this proof is their unboundedness. For a systematic investigation of the properties that the boundaries of cells must be assumed to possess see the paper of Weyl already cited.
That ann ○ element cannot be defined to be anyn-set whose skeleton is annΔelement, and thenproved to be ann-complex, is shewn by the simple example of a 1-cell bounded by a single 0-cell.
cf. H. Kneser, Proc. Amsterdäm27, (1924), p. 601.
H. Kneser,op. cit.. pointed out that this need not be postulated in the definition.
i. e., to eachk-component,U k , of ΔΓ1, there corresponds ak-cell,u k , of\(\bigcirc \mathfrak{s}\Gamma ^1 \) and ifU j is a component ofU k ,u j , boundsu k .
The corresponding notation for arrays was introduced in an earlier paper “On the superposition ofn-dimensional manifolds”, Journal Lond. Math. Soc.2 (1927), p. 56–64, referred to asS. The only modification is that the statement “ΔΓ1 is ΔΓ2” implies that all the pieces in ΔΓ1, aresimplexes.
cf. Weyl,l. c. (Revista di Matem. Hisp-Amer, 1923) The additional transformations allowed by Weyl (his axioms C and D) lead to no increased generality, with the present definitions, in view ofFII Lemmas 7a and 7b (quoted on p. 402.).
This is Theorem 2 ofS.
cf. f. n. 24)i.e., to eachk-component,U k , of ΔΓ1, there corresponds ak-cell,u k , of\(\bigcirc \mathfrak{s}\Gamma ^1 \) and ifU j is a component ofU k ,u j boundsu k . above.
S., Theorem, 2, Case 2.
S., Theorem 3.
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Newman, M.H.A. Topological equivalence of complexes. Math. Ann. 99, 399–412 (1928). https://doi.org/10.1007/BF01459105
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DOI: https://doi.org/10.1007/BF01459105