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Arithmetic Invariants of Simplicial Complexes

Published online by Cambridge University Press:  20 November 2018

M. Brown  and
A. G. Wasserman
M. Brown
Affiliation:
The University of Michigan, Ann Arbor, Michigan
A. G. Wasserman
Affiliation:
The University of Michigan, Ann Arbor, Michigan
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What invariants of a finite simplicial complex K can be computed solely from the values v0(K), V1(K), …, vi(K), … where Vi(K) is the number of i-simplexes of K? The Euler chracteristic χ(K) = Σ i (– 1)ivi(K) is a subdivision invariant and a homotopy invariant while the dimension of K is a subdivision invariant and homeomorphism invariant. In [3], Wall has shown that the Euler chracteristic is the only linear function to the integers that is a subdivision invariant. In this paper we show that the only subdivision invariants (linear or not) of K are the Euler characteristic and the dimension. More precisely we prove the following theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1306 - 1310
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Glaser, L., Geometrical combinatorial topology (Van Nostrand, 1970).Google Scholar
2. Rushing, T. B., Topological embeddings (Academic Press, N.Y., 1973).Google Scholar
3. Wall, C. T. C., Arithmetic properties of simplicial complexes, Can. J. Math. 18 (1966), 92–96.Google Scholar