Before revision:  A map \varphi:G\rightarrow M is a weak embedding if it can be perturbed into an embedding \psi_\varepsilon:G\rightarrow M with \|\varphi-\psi_\varepsilon\|<\varepsilon for every \varepsilon> 0. A polynomial-time algorithm for recognizing weak embeddings was recently found by Fulek and Kyncl , 2017 , which reduces to solving a system of linear equations over Z_2.
After revision:  A map \varphi:G\rightarrow M is a weak embedding if it can be perturbed into an embedding \psi_\varepsilon:G\rightarrow M with \|\varphi-\psi_\varepsilon\|<\varepsilon for every \varepsilon> 0. A polynomial-time algorithm for recognizing weak embeddings was recently found by Fulek and Kyncl , which reduces to solving a system of linear equations over Z_2.