Higher Order Invariants of Immersions of Surfaces into 3-Space

Formulae for order one invariants of immersions and embeddings of surfaces

The universal order 1 invariant f^U of immersions of a closed orientable surface into R^3, whose existence has been established in [N3], takes values in the group G_U = K \oplus Z/2 \oplus Z/2 where K is a countably generated free Abelian group. The projections of f^U to K and to the first and second Z/2 factors are denoted f^K, M, Q respectively. An explicit formula for the value of Q on any embedding has been given in [N2]. In the present work we give an explicit formula for the value of f^K on any immersion, and for the value of M on any embedding.

Invariants of Knots, Embeddings and Immersions via Contact Geometry

This paper is an overview of the idea of using contact geometry to construct invariants of immersions and embeddings. In particular, it discusses how to associate a contact manifold to any manifold and a Legendrian submanifold to an embedding or immersion. We then discuss recent work that creates invariants of immersions and embeddings using the Legendrian contact homology of the associated Legendrian submanifold.

On the Wu invariants for immersions of a graph into the plane

We give an explicit calculation of the Wu invariants for immersions of a finite graph into the plane and classify all generic immersions of a graph into the plane up to regular homotopy by the Wu invariant. This result is a generalization of the fact that two plane curves are regularly homotopic if and only if they have the same rotation number.

Cobordism of immersions of surfaces in non-orientable 3-manifolds

Some properties of non-orientable 3-manifolds are shown. The semi-group of cobordism of immersions of surfaces in such manifolds is computed and proven actually to be a group. Explicit invariants are provided.

Immersions of Non-orientable Surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R^3, with values in any Abelian group. We show they are all functions of the universal order 1 invariant that we construct as T \oplus P \oplus Q where T is a Z valued invariant reflecting the number of triple points of the immersion, and P,Q are Z/2 valued invariants characterized by the property that for any regularly homotopic immersions i,j:F\to R^3, P(i)-P(j) \in Z/2 (respectively Q(i)-Q(j) \in Z/2) is the number mod 2 of tangency points (respectively quadruple points) occurring in an…

Differential 3-knots in 5-space with and without self intersections

Regular homotopy classes of immersions of a 3-sphere in 5-space constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self intersection. Geometric properties of self intersections are used to construct two invariants J and St of generic immersions which are analogous to Arnold's invariants of plane curves. We prove that J and St are independent first order invariants and that any first order invariant is a linear co…

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

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 the 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 th…

Cobordism of fold maps, stably framed manifolds and immersions

We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.

A geometric classification of immersions of 3-manifolds into 5-space

In this paper we define two regular homotopy invariants c and i for immersions of oriented 3-manifolds into R^5 in a geometric manner. The pair (c(f),i(f)) completely describes the regular homotopy class of the immersion f. The invariant i corresponds to the 3-dimensional obstruction that arises from Hirsch-Smale theory and extends the one defined in [10] for immersions with trivial normal bundle.

曲面浸入的Arnold奇异性

Arnold Strangeness of surface immersions

本文定义了一个整数值函数，用于检测和分类闭曲面浸入中涉及四重点与三线相切的跳跃，作为Arnold奇异性不变量的高维类比，并基于共定向数据将四重点跳跃分为五种几何情形。

Immersions of the circle into a surface

We classify immersions $f$ of $S^1$ in a $2$-manifold $M$ in terms of elementary invariants: the parity $S(f)$ of the number of double points of a self-transverse $C^1$-approximation of $f$, and the turning number $T(e\bar f)$ of the immersion $e\bar f:S^1\to M_f\subset\Bbb R^2$, where $\bar f$ is a lift of $f$ to the cover $M_f$ of $M$ corresponding to the subgroup $\left<[f]\right>\subsetπ_1(M)$. Namely, immersions $f,g:S^1\to M$ are regular homotopic if and only if they are homotopic, and if $M=S^2$ or $\Bbb R P^2$ or the normal bundle $ν(f)$ is non-orientable, then $S(f)=S(g)$, whereas if …

曲面不变量的显式切片公式 via 曲线不变量

An explicit slice formula for surface invariants via curve invariants

本文通过曲线不变量提出曲面不变量的显式切片公式，利用运动影片观点引入微分测度，描述曲面不变量在奇点转换中的变化，得出四重点事件中曲面不变量的变化公式。

Immersions of 3-sphere into 4-space associated with Dynkin diagrams of types A and D

We construct two infinite sequences of immersions of the 3-sphere into 4-space, parameterized by the Dynkin diagrams of types A and D. The construction is based on immersions of 4-manifolds obtained as the plumbed immersions along the weighted Dynkin diagrams. We compute their Smale invariants and bordism classes of immersions by using Ekholm-Takase's formula in terms of singular Seifert surfaces. In order to construct singular Seifert surfaces for the immersions, we use the Kirby calculus.

Arnold-type Invariants of Curves on Surfaces

Recently V. Arnold introduced Strangeness and $J^{\pm}$ invariants of generic immersions of an oriented circle to $\R^2$. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface $F$. We explicitly describe all the invariants satisfying axioms, which naturally generalize the axioms used by V. Arnold.

Fold maps and immersions from the viewpoint of cobordism

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by na…

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

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 $F_0$ and $F_8 : T^3 \hookri…

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。

Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant II

The notion of the geometrical $\Z/2 \oplus \Z/2$--control of self-intersection of a skew-framed immersion and the notion of the $\Z/2 \oplus \Z/4$-structure (the cyclic structure) on the self-intersection manifold of a $\D_4$-framed immersion are introduced. It is shown that a skew-framed immersion $f:M^{\frac{3n+q}{4}} \looparrowright \R^n$, $0 < q <<n$ (in the $\frac{3n}{4}+ε$-range) admits a geometrical $\Z/2 \oplus \Z/2$--control if the characteristic class of the skew-framing of this immersion admits a retraction of the order $q$, i.e. there exists a mapping $κ_0: M^{\frac{3n+q}{4}} \to \…

The geometric Hopf invariant and double points

The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in terms of the double point set of f. We interpret the Smale-Hirsch-Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m<2n-1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.

Maximal Sp(4,R) surface group representations, minimal immersions and cyclic surfaces

Let $S$ be a closed surface of genus at least $2$. For each maximal representation $ρ: π_1(S)\rightarrow\mathsf{Sp}(4,\mathbb{R})$ in one of the $2g-3$ exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space $\mathsf{Sp}(4,\mathbb{R})/\mathsf{U}(2)$ is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmüller space. Unlike Labourie's recent results on…

近测地浸入与不连续域

Nearly geodesic immersions and domains of discontinuity

本文定义并研究高阶秩对称空间中的近测地浸入，推广双曲空间主曲率有界浸入，并利用旗流形中的自然纤维化给出某些 Anosov 表示及 $Θ$-正表示的不连续域的显式纤维化。

Cobordism invariants of fold maps

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for -1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallel…

Homotopy type of the complement of an immersion and classification of embeddings of tori

This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\ge p+3q/2+2$, $p\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension.

Geometric approach towards stable homotopy groups of spheres. The Hopf invariant

We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams is obtained in all dimensions except 15 and 31. To prove that the stable Hopf invariant H: Π_n \to Z/2 vanishes for n>31, we apply methods of geometric topology. The Pontrjagin-Thom construction along with Hirsch's compression lemma identify every α\in Π_n with the framed bordism class of a framed immersion of a closed n-manifold into R^{n+k}, for any given k>0. Its self-intersection M projects to an immersion f: M …

Lin-Wang type formula for the Haefliger invariant

In this paper we study the Haefliger invariant for long embeddings $\mathbb{R}^{4k-1}\hookrightarrow\mathbb{R}^{6k}$ in terms of the self-intersections of their projections to $\mathbb{R}^{6k-1}$, under the condition that the projection is a generic long immersion $\mathbb{R}^{4k-1}\looparrowright\mathbb{R}^{6k-1}$. We define the notion of "crossing changes" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in $\mathbb{R}^{4k-1}$. This formula is a higher-dimensional analogue to t…

Embeddings, Normal Invariants and Functor Calculus

This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie's homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

The Hopf invariant of a Haefliger knot

We show that Haefliger's differentiable (6,3)-knot bounds, in 6-space, a 4-manifold (a Seifert surface) of arbitrarily prescribed signature. This implies, according to our previous paper, that the Seifert surface has been prolonged in a prescribed direction near its boundary. This aspect enables us to understand a resemblance between Ekholm-Szucs' formula for the Smale invariant and Boechat-Haefliger's formula for Haefliger knots. As a consequence, we show that an immersion of the 3-sphere in 5-space can be regularly homotoped to the projection of an embedding in 6-space if and only if its Sma…

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…

Configuration spaces and Vassiliev classes in any dimension

The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.

Arnold-type invariants of wave fronts on surfaces

Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface $F$. All invariants satisfying the axioms which naturally generalize the axioms used by Arnold are explicitly described. We also give an explicit formula for the finest order one $J^+$-type invariant of fronts on an orientable surface $F\neq S^2$. We obtain necessary and sufficient conditions for an invariant of nongeneric fronts with one nongeneric singular point to be the Va…