Smooth p-adic analytic spaces are locally contractible

On Non-Commutative Analytic Spaces Over Non-Archimedean Fields

“…Analytic spaces in the sense of Berkovich have better local properties (e.g. they are locally arcwise connected, see [Be3], while in the classical rigid analytic geometry the topology is totally disconnected).…”

“…They are based on the version of Tate algebra in which the commutativity of variables z i z j = z j z i is replaced by the q-commutativity z i z j = qz j z i , i < j, q ∈ K × , |q| = 1. In particular our non-commutative analytic K3 surface is defined as a ringed space, with the underlying topological space being an ordinary K3 surface equipped with the natural Grothendieck topology introduced in [Be3] and the sheaf of noncommutative algebras which is locally isomorphic to a quotient of the abovementioned "quantum" Tate algebra. The construction of a quantum K3 surface uses a non-commutative analog of the map π.…”

“…Skeleta can be defined for more general analytic spaces (see [Be3]). For example, the skeleton of the d-dimensional Drinfeld upper-half space Ω d K is the Bruhat-Tits building of P GL(d, K).…”

Degenerating Families of Dendrograms

“…The new points corresponding to (2), (3) or (4) The concept of generic Berkovich points via multiplicative seminorms works also in higher dimension, and the result is that p-adic manifolds are locally contractible [4]. In any case, by that concept, the data domain can be viewed as a contiunuum.…”

Sur Les Composantes Connexes D’une Famille D’espaces Analytiques -Adiques

“…Poineau 2 par exemple, depuis [5], que ces espaces sont localement contractiles. Ce résultat a été étendu aux analytifiées de variétés quasi projectives par E. Hrushovski et F. Loeser, via l'utilisation de la théorie des modèles.…”