\partial \Omega}$ and also of the possible multi-sheetedness of the intermediate stepwise extensions. This causes considerably more delicate topological problems than in the well known Grauert bump method, in which monodromy of the holomorphic (or meromorphic, or sheaf-theoretic) extensions from $\{ a < \rho < b \}$ to $\{ a' < \rho < b \}$ with $a
' < a$ is almost freely assured[^1], even across critical points of $\rho$. Considering simply a domain $\Omega \Subset {\mathbb{C}}^n$ ($n\geqslant 2$), with obvious exhaustion $\rho (z) := \vert \! \vert z \vert \! \vert$, the classical Hartogs theorem based on analytic discs and on Morse theory was worked out in [@mp2006b], where emphasis was put on rigor in order to provide with firm grounds the subsequent works on the subject. The essence of the present article is to transfer such an approach to $(n - 1)$-complete general complex spaces, where $\overline{ \partial }$ techniques are still lacking, with some new difficulties due to the singularities.

§2. Statement of the results {#statement-of-the-results .unnumbered}
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Thus, let $\big( X,\mathcal{ O}_X \big)$ be a reduced complex analytic space of pure dimension $n \geqslant 2$, equipped with an open cover $X = \bigcup_{ j\in J}\, U_j$ 