Gluing instantons \`a la Brezis-Coron in dimension four and the dipole construction

Given a connection $A$ on a $SU(2)$-bundle $P$ over $\mathbb{R}^4$ with finite Yang-Mills energy $YM(A)$ and nonzero curvature $F_A(0)$ at the origin, and given $\rho>0$ small enough, we construct a new connection $\hat A$ on a bundle $\hat P$ of different Chern class ($|c_2(A)-c_2(\hat A)|=8\pi^2$), in such a way that $\hat A$ is gauge equivalent to $A$ in $\mathbb{R}^4\setminus B_\rho(0)$, gauge equivalent to an instanton in a smaller ball $B_{\tau \rho}(0)$, and $$YM(\hat A)\le YM(A)+8\pi^2-\varepsilon_0\rho^4|F_A(0)|^2,$$ where $\tau\in (0.3,0.4)$ and $\varepsilon_0>0$ are universal constant independent of $A$ and $\rho$. Our gluing method is similar in spirit to the one of Brezis-Coron for harmonic maps. We compare it with classical results by Taubes and discuss applications and open problems.