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00925njm 22002777a 4500 dc Chapuy, G. author Perarnau Llobet, Guillem author 2020-03-11 This is a post-peer-review, pre-copyedit version of an article published in Discrete and computational geometry. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00454-020-00189-w We give superexponential lower and upper bounds on the number of coloured d-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and d=3 is fixed. In the special case of dimension 3, the lower and upper bounds match up to exponential factors, and we show that there are 2O(n)nn6 coloured triangulations of 3-manifolds with n tetrahedra. Our results also imply that random coloured triangulations of 3-manifolds have a sublinear number of vertices. The upper bounds apply in particular to coloured d-spheres for which they seem to be the best known bounds in any dimension d=3, even though it is often conjectured that exponential bounds hold in this case. We also ask a related question on regular edge-coloured graphs having the property that each 3-coloured component is planar, which is of independent interest. This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. ERC-2016-STG 716083 “CombiTop”). G. Perarnau acknowledges an invitation in Paris funded by the ERC Grant CombiTop, during which this project was advanced. Postprint (author's final draft) Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria Combinatorial analysis Triangulated manifolds Random complexes Enumeration Anàlisi combinatòria On the number of coloured triangulations of d-manifolds