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- W2000546249 abstract "1. In his first memoir on analysis situst Poincaredefined a number N(Fk, Jin_k) which had previously been considered, at least in special cases, by Kronecker. With certain conventions as to sign this number represents the excess of the number of positive over the number of negative intersections of a k-dimensional circuit Fk with an (n k) -dimensional circuit F,i_ when both are immersed in an n-dimensional oriented manifold. The purpose of the present paper is to show how to calculate this number when the manifold is defined combinatorially as a collection of cells and the circuits are composed of sets of these cells; and to show how the matrices which represent the inter. sectional relations between the k-circuits and the (n k) circuits depend on the matrices of orientation of the manifold. We also define certain modulo 2 intersection numbers and discuss the matrices coinnected with them. The terminology and notations of the Cambridge Colloquittm Lectures on Analysis Situs (New York, 1922) will be used without further explanation, and the references not otherwise indicated will be to that book. 2. Let a manifold Mn be given as the set of all points of a complex C(. Let C' be a complex dual to Cn constructed as explained on page 88 by means of a complex Cn which is a regular subdivision both of Cn aind of C'. Every k-cell 0 of Cn has a single point PI (cf. p. 85) in common with a single (n -k)-cell of C' whiclh is called bjy-k. Our first problem will be to assign a positive or negative sign to the intersection of 0 with b.k. In order to do this, we suppose M, to be orienited as explained in Chapter IV and that all cells, circuits,. etc., are oriented. Moreover, in the regular complex Cn, in which each i-cell is uniquely determined by its i + 1 vertices, the orientation of the i-cell will be denoted by the order in which its vertices are written, and the following two conventions will be followed: (1) if AO A1 .. Ak denotes a given oriented k-cell (k 1, 2, ..., n) any even permutation of Ao A1 .. Ak denotes the same oriented k-cell and any odd permutation denotes its negative; (2) the oriented (k 1 )-cell A1 A2 ... Ak iS positively related to the oriented k-cell Ao A, * * Ak." @default.
- W2000546249 created "2016-06-24" @default.
- W2000546249 creator A5054558618 @default.
- W2000546249 date "1923-04-01" @default.
- W2000546249 modified "2023-09-24" @default.
- W2000546249 title "The intersection numbers" @default.
- W2000546249 doi "https://doi.org/10.1090/s0002-9947-1923-1501259-6" @default.
- W2000546249 hasPublicationYear "1923" @default.
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