Correction to Our Article “Topology of Random 2Complexes” Published in DCG 47 (2012), pp. 117–149
Correction to Our Article “Topology of Random
D. Cohen 0 1 2
A. Costa 0 1 2
M. Farber 0 1 2
T. Kappeler 0 1 2
A. Costa 0 1 2
0 Mathematical Institute, University of Zurich , Winterthurerstrasse 190, 8057 Zurich , Switzerland
1 School of Mathematical Sciences, Queen Mary University of London , London E1 4NS , UK
2 Department of Mathematics, Louisiana State University , Baton Rouge, LA 70803 , USA
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.

On the contrary, any closed triangulated surface S with negative Euler
characteristic, χ (S) < 0, admits a subdivision which is unbalanced. Indeed, let S be a triangulated
implying that
closed surface with χ (S) < 0 and let X be obtained from S by removing the interior
of a single 2simplex σ . Then
v( X ) = v(S) = f (S)/2 + χ (S),
f ( X ) = f (S) − 1
μ(S) < μ( X ) < 1/2,
(
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.
μ(S ) =
v(S) + k
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 )