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Smooth $$p$$-adic analytic spaces are locally contractible. II. (English) Zbl 1060.32010
Adolphson, Alan (ed.) et al., Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter (ISBN 3-11-017478-2/hbk). 293-370 (2004).
The paper is a continuation of the author’s article [V. G. Berkovich, Invent. Math. 137, 1–84 (1999; Zbl 0930.32016)]. Let $$k$$ be a complete non-Archimedean nontrivially valued field. The author considers smooth $$k$$-analytic spaces [see V. G. Berkovich, Publ. Math. Inst. Hautes Étud. Sci. 78, 5–161 (1993; Zbl 0804.32019)]. It was shown in the first mentioned paper that any strictly analytic subdomain $$X$$ of a smooth $$k$$-analytic space is locally contractible. The paper under review is devoted mainly to the proof of the following stronger result.
Each point $$x\in X$$ has a fundamental system of open neighbourhoods $$V$$, such that: (a) there is a contraction $$\Phi$$ of $$V$$ to a point $$x_0\in V$$; (b) there is an increasing sequence of compact strictly analytic domains $$X_1\subset X_2\subset \cdots \subset V$$ which exhaust $$V$$ and are preserved under $$\Phi$$; (c) for any bigger non-Archimedean field $$K$$, $$V\widehat \otimes K$$ has a finite number of connected components and $$\Phi$$ lifts to a contraction of each of them to a point over $$x_0$$; (d) there is a finite separable extension $$L$$ of $$k$$, such that if the above $$K$$ contains $$L$$, then the map $$V\widehat \otimes K\to V\widehat \otimes L$$ induces a bijection between the sets of connected components.
In addition, the author develops, for general $$k$$-analytic spaces, in the case of zero characteristic, a Coleman type of $$p$$-adic integration theory. For the case of smooth $$k$$-analytic curves, see R. Coleman [Invent. Math. 69, 171–208 (1982; Zbl 0516.12017)] and R. Coleman and E. de Shalit [Invent. Math. 93, 239–266 (1988; Zbl 0655.14010)].
For the entire collection see [Zbl 1047.14001].

##### MSC:
 32P05 Non-Archimedean analysis 32C35 Analytic sheaves and cohomology groups