Isotopy Invaraints of Immersed surfaces in a 4-manifold

Abstract

In this this dissertation we introduce an isotopy invariant of generically immersed surfaces in some 4-manifold. The construction is based on Khovanov homology and its variants in the same way as the construction of Turaev-Viro module of a 3-manifold with infinite cyclic covering relies on TQFT. The invariant is first constructed for generically immersed surfaces in S<super>3</super> &times S<super>1</super> using the functoriality of Khovanov homology, and is generalized by using new versions of Khovanov homology. Moreover, it is also generalized to surfaces generically immersed transversal to a standardly embedded S<super>2</super> in S<super>4</super>. Examples are studied to illustrate the strength and weakness of this invariant.