Constructing Immersions from Three-Manifolds to Four-Dimensional Space

Abstract

The aim of this thesis is to construct codimension one immersions of surfaces and 3-manifolds to compute invariants of their regular homotopy classes. It is known that every orientable closed 3-manifold is represented by a framed link L in.S3. We can modify L to be an even framed link by Kirby's moves [B-P]. The even framing of L allows us to construct an immersed 3-manifold in R4 using tracks over the boundaries of a tubular neighbourhood of L in S3. Each track is realised as a regular homotopy between two embedded tori. A regular homotopy can be described by a sequence of geometric local deformations [H-N1][H-N2]. We find explicit expressions of deformations of double curves in an immersion of F2 (see Section 3.4.7). There is a one-to-one correspondence between the set of regular homotopy classes of immersions of a surface .F2 in R3 and H1(F2;Z2). We describe the one-to-one correspondence geometrically using deformations related to twisting annuli. This gives an interpretation of a class of H1(F2;Z2) as an immersed annulus in immersed surface F2 in R3. For this description we provide a construction called a bug construction whose data is a loop on the surface F2 in R3. We prove that for a given even framed link representation of a 3-manifold M3, we can construct an immersion, in which some invariants can be computed. We obtain two different types of tracks for the constructions. This enables us to compute these invariants. We prove that a regular homotopy class of immersed 3-manifold in R4 can be constructed by tracks of tori. This implies that the regular homotopy class is determined by the geometric complexity produced by twistings of the tracks.