*marking data* $\mu$ on a bifold $\check X$, we mean a pair $(U_{\mu}, E_{\mu})$ consisting of an open set $U_{\mu}$ and an $\SO(3)$ bundle $E_{\mu} \to U_{\mu}\cap \check X^{o}$ (where $\check
X^{o}$ is the locus of non-orbifold points). A *marked* bifold connection is a bifold connection $(E,A)$ on $\check X$ together with a choice of an equivalence class of an isomorphism $\sigma$ from $E|_{U_{\mu}\cap \check X^{o}}$ to $E_{\mu}|_{U_{\mu}\cap\check
  X^{o}}$. Two isomorphism $\sigma_{1}$ and $\sigma_{2}$ are equivalent if $\sigma_{1}\comp\sigma_{2}^{-1} : E_{\mu}\to E_{\mu}$ lifts to the determinant-$1$ gauge group, i.e. a section of the associated bundle with fiber $\SU(2)$. The marking data is *strong* if the automorphism group of every $\mu$-marked bifold connection is trivial. In dimension $3$, a sufficient condition for $\mu$ to be strong is that $U_{\mu}$ contains a point $x$ with $H_{x}=V$ (a vertex of the corresponding web), or that $U_{\mu}\check
X^{o}$ contains a torus on which $w_{2}(E_{\mu})$ is non-zero.

#### Instanton homology.

Let $\check Y$ be a closed, connected, oriented $3$-dimensional bifold with strong marking data $\mu$. The set of isomorphism classes of $\mu$-marked bifold connections of Sobolev class $L^{2}_{k}$, for large enough $k$, is parametrized by a Hilbert manifold $\bonf_{k}(\check Y;\mu)$. Using the perturbed Chern-Simons functional, one 