Embeddings of Surfaces into 3-Space and Quadruple Points of Regular Homotopies

Subgroups of the Mapping Class Group and Quadruple Points of Regular Homotopies

Let F be a closed orientable surface. If i,i':F \to R^3 are two regularly homotopic generic immersions, then it has been shown in [N] that all generic regular homotopies between i and i' have the same number mod 2 of quadruple points. We denote this number by Q(i,i') \in Z/2. We show that for any generic immersion i:F\to R^3 and any diffeomorphism h:F\to F such that i and i\circ h are regularly homotopic, Q(i,i\circ h) = (rank(h_*-Id) + (n+1)e(h)) mod 2, where h_* is the map induced by h on H_1(F,Z/2), n is the genus of F and e(h) is 0 or 1 according to whether h is orientation preserving or r…

Quadruple Points of Regular Homotopies of Surfaces in 3-Manifolds

Let GI denote the space of all generic immersions of a surface F into a 3-manifold M. Let q(H_t) denote the number mod 2 of quadruple points of a generic regular homotopy H_t : F -> M. We are interested in defining an invariant Q : GI -> Z/2 such that q(H_t) = Q(H_0) - Q(H_1) for any generic regular homotopy H_t : F -> M. Such an invariant exists iff q=0 for any "closed" generic regular homotopy (abbreviated CGRH.) Max and Banchoff proved that for any CGRH H_t: S^2 -> R^3 indeed q(H_t)=0. We generalize their result as follows: Theorem 3.9: Let F be a system of closed surfaces, and let H_t : F …

Positive Legendrian regular homotopies

In contrast with what happens for Legendrian embeddings, there always exist positive loops of Legendrian immersions.

无三重点浸入球面的不变量

An Invariant for Triple-Point-Free Immersed Spheres

本文定义了2-球面到欧几里得3-空间的无三重点浸入在正则同伦下保持不变的不变量，并证明该空间有无穷多个正则同伦类。

There are non homotopic framed homotopies of long knots

Let $\mathcal {M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl(Σ^{(1)}_{iness})$ be the closure of the union of all singular knots in $\mathcal {M}$ with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call $Σ^{(1)}_{iness}$ the {\em inessential walls} and we call $\mathcal {M}_{ess} = \mathcal {M} \setminus cl(Σ^{(1)}_{iness})$ the {\em essential diagram space}. We construct a non trivial class in $H^1(\mathcal {M}_{ess}; \mathbb{Z}[A,…

Integrating curvature: from Umlaufsatz to J^+ invariant

Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies. A natural question is whether one can find some non-trivial densities on a curve, such that the corresponding integrals are (possibly after some corrections) also invariant under regular homotopies of the curve in the class of generic immersions. We construct a family of such densities using indices of points relative to the curve. This family depends on a f…