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Intersection-complexes. I. Combinatory Theory

Published online by Cambridge University Press:  24 October 2008

M. H. A. Newman
M. H. A. Newman
Affiliation:
St John's College
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Extract

The theory of the intersection-complex Ch · Ck, due originally to Veblen, Weyl, Alexander and Lefschetz, has recently been extended by Flexner to cover intersections on a “topological n-manifold” (locally Cartesian n-space) in the case h + k = n. In his theory the whole of Lefschetz's work for ordinary simplicial manifolds is presupposed, including the rather difficult approximations. The object of the present paper is to provide a combinatory theory of the general intersection, (h + k ≥ n), which would serve equally well as a basis for Flexner's work, and avoids the wasteful process of using two independent approximations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 4 , October 1931 , pp. 491 - 501
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* Flexner, W. W., Annals of Math. (2), 32 (1931), 393–406.CrossRefGoogle Scholar

† Annals of Math. 31 (1930), p. 294.Google Scholar

‡ The product C hC k, an (h + k + 1)-chain, is not, of course, to be confused with the intersection C h · C k, to be defined later.

* Cf. Alexander, op. cit.; Newman, , Journal Lond. Math. Soc. 6 (1931), 186–192.CrossRefGoogle Scholar

† I.e. to each component of A n corresponds a vertex of 1A n, and to each rising sequence of components, A 0, A 1,…, A n, of A n (A iCA i+1), corresponds an n-component, β0 β1 … βn, of 1A n. The indicatrix is fixed thus: Each A i contains one vertex, a i, not in A i−1. If

then εβ0 β1 … βn is the corresponding component of 1A n.

‡ Two chains are isomorphic if they are the same functions of the same or different “marks.”

† Lefschetz gives the two symbols C h · C n−h and (C h · C n−h) distinct meanings. Although this notation has been followed by other writers the inconvenience of not being able to use brackets in their ordinary punctuating sense is so great that I have ventured to give up the distinction. Moreover, in the present paper no special definition is necessary, since Lefschetz's (C h · C n−h) appears naturally as .