Israel Journal of Mathematics

, Volume 45, Issue 4, pp 337–351 | Cite as

Enumeration ofQ-acyclic simplicial complexes

  • Gil Kalai


(n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely ( k n−1 )k-faces, (3)H k (C)=0. We prove that for
,Hk−1(C) is a finite group, and our main result is:
. This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.


Finite Group Simplicial Complex Incidence Matrix Algebraic Topology Free Abelian Group 
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Copyright information

© Hebrew University 1983

Authors and Affiliations

  • Gil Kalai
    • 1
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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