English

Abstract:
Topologists studying 3-dimensional manifolds have been interested in finding $\pi_{1}$-injective surfaces immersed in a 3-manifold and trying to lift the immersions to embeddings in some finite-degree covering space of the manifold. Such an approach was definitely helpful in understanding surface groups, and Haken, Waldhausen, etc. have proved that this approach is also essential in understanding the topology of 3-manifolds and their fundamental groups. We introduce related group-theoretical concepts such as ERF, LERF, and RF groups and explain how these properties are associated with "virtually embedded" surfaces immersed in 3-manifolds. We particularly see why they are important in light of conjectures by Waldhausen, Thurston, and others. We then survey the historical development of this problem, including earlier conjectures, counterexamples, some essential theorems, and current conjectures. Finally we look at a few recent results by Neumann, Gitik, Niblo, Wise, etc. on attempts to find and lift surfaces immersed in 3-manifolds of various types, such as knot complements and graph manifolds.