Black hole and wormhole branches in gravitational decoupling
Minimal Geometric Deformation (MGD) applied to a static Schwarzschild black hole seed generates a single decoupler function $h(r)$, obtained by solving the $θ$-sector field equations together with an equation of state. Once $h(r)$ is fixed, the resulting one-parameter family is controlled by the coupling strength $k$ through $F(r;k)=1+k\,h(r)$. We show that, whenever the deformation develops a simple outermost root that crosses the seed horizon, the same fixed decoupler leads to two mutually exclusive branches associated with different global completions: on one side of the critical coupling the deformed metric preserves the seed horizon as a black hole, whereas on the other side the root $r_*>2M$ lies in the exterior and cannot be interpreted as an interior modification of the black hole geometry. We prove that this root forces a loss of Lorentzian signature on the interval $(2M,r_*)$, so that no smooth extension of the exterior metric through the seed horizon $r=2M$ exists once $r_*$ lies outside it. Within the static, spherically symmetric class considered here, the corresponding smooth Lorentzian completion is a two-ended wormhole obtained by excising $(2M,r_*)$ and doubling the region $r\geq r_*$ across the minimal sphere $\mathcal T=\{r=r_*\}$. No topology change of any single spacetime is claimed or required: $k>k_c$ and $k<k_c$ simply correspond to two different, non-diffeomorphic manifolds, and Lemma~1 below shows that the metric itself dictates which of the two is the admissible completion for a given $k$. We compute the second homology group of both completions explicitly, $H_2(Σ_{\rm BH},\mathcal H)=0$ for the black hole exterior relative to its horizon and $H_2(Σ_{\rm WH})\cong\mathbb Z$ for the completed wormhole manifold, giving a discrete invariant that distinguishes the two branches.