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Arithmetic Invariants of Subdivision of Complexes

  • C. T. C. Wall (a1)

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The following problem was raised by M. Brown. Let K be a finite simplicial complex, of dimension n, with αi(K) simplexes of dimension i. Which of the linear combinations have the property that they are unaltered by all stellar subdivisions of K? The most obvious invariant is the Euler characteristic; there are also some identities that hold for manifolds (2), so, if K is a manifold, they remain true on subdivision. We shall see that no other expressions are ever invariant, but if K resembles a manifold in codimensions ⩽2r (in a sense defined below) that r of the relations continue to hold.

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References

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1. Alexander, J. W., The combinatorial theory of complexes, Ann. of Math., 81 (1930), 292320.
2. Klee, V., A combinatorial analogue of Poincaré's duality theorem, Can. J. Math., 16 (1964), 517531.
3. Seifert, H. and Threlfall, W., Lehrbuch der Topologie (Leipzig, 1934).
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