Let $Y$ be a smooth compact $n$-manifold. We study smooth embeddings and immersions $β: M \to \mathbb R \times Y$ of compact $n$-manifolds $M$ such that $β(M)$ avoids some a priory chosen closed poset $Θ$ of {\sf tangent patterns} to the fibers of the obvious projection $π: \mathbb R \times Y \to Y$. Then, for a fixed $Y$, we introduce an equivalence relation between such $β$'s; it is a crossover between pseudo-isotopies and bordisms. We call this relation {\sf quasitopy}. In the study of quasitopies, the spaces $\mathcal P_d^{\mathbf cΘ}$ of real univariate polynomials of degree $d$ with real divisors, whose combinatorial patterns avoid a given closed poset $Θ$, play the classical role of Grassmanians. We compute the quasitopy classes $\mathcal{QT}_d^{\mathsf{emb}}(Y, \mathbf cΘ)$ of $Θ$-constrained embeddings $β$ in terms of homotopy/homology theory of spaces $Y$ and $\mathcal P_d^{\mathbf cΘ}$. We prove also that the quasitopies of emeddings stabilize, as $d \to \infty$.
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.
Let M and N be smooth manifolds. For an open V of M let emb(V,N) be the space of embeddings from V to N. By results of Goodwillie and Goodwillie-Klein, the cofunctor V |--> emb(V,N) is analytic if dim(N)-dim(M) > 2. We deduce that its Taylor series converges to it. For details about the Taylor series, see Part I.
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.
We show that a compact orientable 4-manifold M has a CR regular immersion into C3 if and only if both its first Pontryagin class and its Euler characteristic vanish, and has a CR regular embedding into C3 if and only if in addition the second Stiefel-Whitney class of M vanishes.
Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M,N) should come from an analysis of the cofunctor V |--> emb(V,N) from the poset O of open subsets of M to spaces. We therefore abstract some of the properties of this cofunctor, and develop a suitable calculus of such cofunctors, Goodwillie style, with Taylor series and so on. The terms of the Taylor series for the cofunctor V |--> emb(V,N) are explicitly determined. In a sequel to this paper, we introduce the concept of an analytic cofunctor from O to 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…
We study {\sf traversing} vector flows $v$ on smooth compact manifolds $X$ with boundary. For a given compact manifold $\hat X$, equipped with a traversing vector field $\hat v$ which is {\sf convex} with respect to $\partial\hat X$, we consider submersions/embeddings $α: X \to \hat X$ such that $\dim X = \dim \hat X$ and $α(\partial X)$ avoids a priory chosen tangency patterns $Θ$ to the $\hat v$-trajectories. In particular, for each $\hat v$-trajectory $\hatγ$, we restrict the cardinality of $\hatγ\cap α(\partial X)$ by an even number $d$. We call $(\hat X, \hat v)$ a {\sf convex pseudo-enve…
We show that for the spaces of spherical embeddings modulo immersions $\bar{Emb}(S^n,S^{n+q})$ and long embeddings modulo immersions $\bar{Emb}_{\partial}(D^n,D^{n+q})$, the set of connected components is isomorphic to $π_{n+1}(SG,SG_q)$ for $q\geq 3$. As a consequence, we show that all the terms of the long exact sequence of the triad $(SG;SO,SG_q)$ have a geometric meaning relating to spherical embeddings and immersions.
An immersion of a smooth $n$-dimensional manifold $M \to \mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y \in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the minimum dimension $TN(M)$ such that there exists a totally nonparallel immersion $M \to \mathbb{R}^{TN(M)}$. In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion problem for real projective spaces, and nonsingular symmetric b…
This article presents an improvement and extension of the heuristic first presented by Hougardy, Lutz, and Zelke in 2010 for realizing triangulated orientable surfaces with few vertices by a simplex-wise linear embedding. The improvement consists in the applicability to non-orientable surfaces (simplex-wise linear immersions) as well as symmetric realizations. With the help of our algorithm, numerous - often symmetric - realizations of non-orientable surfaces with the minimal number or few vertices have been obtained for the first time. These examples include the the Projective Plane with one …
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.
A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free …
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 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…
We give a complete obstruction to turning an immersion of an m-dimensional manifold M in Euclidean n-space into an embedding when 3n>4m+4. It is a secondary obstruction, and exists only when the primary obstruction, due to Haefliger, vanishes. The obstruction lives in a twisted cobordism group, and its vanishing implies the existence of an embedding in the regular homotopy class of the given immersion in the range indicated. We use Goodwillie's calculus of functors, following Weiss, to help organize and prove the result.
Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical projection induces the branched covering map. In the case of the 3-sphere, the branch locus is a knotted or linked curve in space, the 2-fold branched cover always embeds, and the 3-fold branch cover might be immersed. In the case of the 4-sphere, the branch locus is a knotted or link…
In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate classes to other maps of other appropriate classes are fundamental and important. In this paper, we consider Morse functions such that inverse images of regular values are disjoint unions of spheres, which are extensions of Morse functions with just two singular points on homotop…
We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of connected components of those embedding spaces through combinatorially defined $L_\infty$-algebras of diagrams.
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.
We give examples of finite, simplicial $2$-complexes that do not PL embed in $\mathbb{R}^4$ and exhibit, for each such complex, a family of PL immersions into $\mathbb{R}^4$ that hide the obstruction to embedding in "higher and higher order Milnor invariants". In addition, we show that the embedding obstructions defined by Krushkal in [8] vanish for our examples. We also answer a question in a paper of Avramidi-Okun-Schreve.
A stable smooth map $f:N\to M$ is called "$k$-realizable" if its composition with the inclusion $M\subset M\times\Bbb R^k$ is $C^0$-approximable by smooth embeddings; and a "$k$-prem" if the same composition is $C^\infty$-approximable by smooth embeddings, or equivalently if $f$ lifts vertically to a smooth embedding $N\to M\times\Bbb R^k$. It is obvious that if $f$ is a $k$-prem, then it is $k$-realizable. We refute the long-standing conjecture that the converse is always true. Namely, for each $n=4k+3\ge 15$ there exists a stable smooth immersion $S^n\to\Bbb R^{2n-7}$ that is $3$-realizable …
We prove that any compact almost complex manifold $(M, J)$ of real dimension $2m$ admits a pseudo-holomorphic embedding in a Euclidean space of dimension $4m + 2$, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class of $(M, J)$, for the existence of an embedding or an immersion in an almost complex Euclidean $4m$-space. We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in $R^6$.
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…
The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give a delooping of the long embedding space, which can be regarded as a version of Morlet-Burghelea-Lashof's delooping of the diffeomorphism group of the disk relative to the boundary. As a corollary, we show that the homotopy fiber is weakly equivalent to a space on which the frame…
This is an investigation into a classification of embeddings of a surface in Euclidean $3$-space. Specifically, we consider $\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $π:\mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}^2$ be the natural projection map onto the Euclidean plane. Let $ \varepsilon : S_g \hookrightarrow \mathbb{R}^2 \times \mathbb{R}$ be a smooth embedding of a closed oriented genus $g$ surface such that the set of critical points for the map $π\circ \varepsilon$ is a smooth (possibly multi-component) $1$-manifold, $\mathscr{C} \subset S_g…
Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions, fold maps and more general generic maps. It is one of important and interesting branches of the singularity theory of differentiable maps and applications to geometry of manifolds. In this paper we present fold maps with information of cohomology rings of their Reeb spaces. Reeb spaces are defined as the spaces of all connected components of all preimages, and …
