Correction to Our Article “Topology of Random 2Complexes” Published in DCG 47 (2012), pp. 117–149
Discrete Comput Geom
Correction to Our Article “Topology of Random 2Complexes” Published in DCG 47 (2012), pp. 117149
D. Cohen 0 1 2
A. Costa 0 1 2
M. Farber 0 1 2
T. Kappeler 0 1 2
0 School of Mathematical Sciences, Queen Mary University of London , London E1 4NS , UK
1 Department of Mathematics, Louisiana State University , Baton Rouge, LA 70803 , USA
2 Mathematical Institute, University of Zurich , Winterthurerstrasse 190, 8057 Zurich , Switzerland
Theorem 27 from our article [1] states that any closed triangulated surface S is balanced, i.e. μ( X ) ≥ μ(S) for any subcomplex X ⊂ S. Here the notation μ(Y ) stands for the ratio v/ f where v and f are the numbers of vertices and faces (i.e. 2simplexes) in a 2complex Y correspondingly. We noticed that the arguments of the proof of Theorem 27 are valid only under an additional assumption that χ (S) ≥ 0. The assumption χ (S) ≥ 0 is implicitly used in the sentence “Since f ≥ f the above inequality follows from 2 − 2b1(S ) + e0 ≥ 4 − 4g” on page 134. Thus, Theorem 27 from [1] should read:
Any closed connected triangulated surface S with χ (S) ≥ 0 is balanced

closed surface with χ (S) < 0 and let X be obtained from S by removing the interior
of a single 2simplex σ . Then
f ( X ) = f (S) − 1
since χ (S) < −1/2. Let S be obtained from S by subdividing the simplex σ and
introducing k new interior vertices in the interior of σ . Then
v(S ) = v(S) + k,
f (S ) = f ( X ) + 2k + 1 = f (S) + 2k.
is approaching 1/2 for k → ∞. Thus we may find k large enough so that μ(S ) >
μ( X ), in view of (1). This shows that S is unbalanced since X is a subcomplex of S .
This correction affects only statement 5 of Corollary 28 from [1] which should be
reformulated as follows:
If p n−1/2+ε for some ε > 0 then, given a topological type of a closed surface,
there exists f0 = f0(ε) such that any balanced triangulation of the surface having
more than f0 2simplexes will be simplicially embeddable into a random 2complex
Y , a.a.s. In particular, if p n−1/2+ε, a random 2complex Y contains small closed
orientable and nonorientable surfaces of all topological types, a.a.s.
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1. Cohen , D. , Costa , A. , Farber , M. , Kappeler , T.: Topology of random 2complexes . Discrete Comput. Geom . 47 , 117  149 ( 2012 )