Masamichi Takase

Regular homotopy classes of immersions of 3-manifolds into 5-space

Abstract We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings

HOMOLOGY 3-SPHERES IN CODIMENSION THREE

F_0

Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions

and of the 3-torus T3 with the following properties: (1) is regularly homotopic to F8 for some immersion , and (2) the immersion h as above cannot be chosen from a regular homotopy class containing an embedding.

Spin structures and codimension two embeddings of 3-manifolds up to regular homotopy

For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers. It also gives rise to a complete invariant for homology bordism classes of all embeddings of homology 3-spheres in the 6-sphere. As a consequence, we show that two embeddings of an oriented integral homology 3-sphere in the 6-sphere are isotopic if and only if they are homology bordant. We also relate our invariant to the Rohlin invariant and accordingly characterize those embeddings which are compressible into the 5-sphere.

Knots and links of complex tangents

A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self-intersection points equal to-n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing these circle bundle immersions with their universal covering maps, we get for n > 0 immersions g(n) of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g(n). The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g(n). As an application, we determine the classes represented by g(n) in the cobordism group of immersions which is naturally identified with the stable 3-stem. It follows in particular that g(n) represents a generator of the stable 3-stem if and only if n is divisible by 3.

Generic immersions and totally real embeddings

We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into 5-space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of 3-manifolds M 3 , the following phenomenon occurs: there exists a codimension two immersion of the 3-sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of M 3 . This involves introducing and studying an equivalence relation on the set of spin structures on M 3 . Their associated μ-invariants also play an important role.

On a move reducing the genus of a knot diagram

It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised as the set of complex tangents of a smooth embedding of the 3-manifold into the three-dimensional complex space if and only if it represents the trivial integral homology class in the 3-manifold. The proof involves a new application of singularity theory of differentiable maps.

An Ekholm–Szűcs-type formula for codimension one immersions of 3-manifolds up to bordism

We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed orientable (4k+1)-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4k-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.

Regular-equivalence of 2-knot diagrams and sphere eversions

For a knot diagram we introduce an operation which does not increase the genus of the diagram and does not change its representing knot type. We also describe a condition for this operation to certainly decrease the genus. The proof involves the study of a relation between the genus of a virtual knot diagram and the genus of a knotoid diagram, the former of which has been introduced by Stoimenow, Tchernov and Vdovina, and the latter by Turaev recently. Our operation has a simple interpretation in terms of Gauss codes and hence can easily be computer-implemented.