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.
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.
Immersions of graphs to the projective plane are studied. A classification of immersions up to regular homotopy is given. A complete invariant of immersions up to regular homotopy is constructed. Equivalence classes are described.
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…
In this paper we generalize the notion of regular homotopy of immersions of a closed connected n-manifold into R^{2n-1} to locally generic mappings. The main result is that if n=2 then two mappings with singularities are regularly homotopic if and only if they have the same number of cross-cap (or Whitney-umbrella) singularities. As an application, we get a description of the path-components of the space of those immersions of a surface into R^4 whose projections into R^3 are locally generic.
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 \…
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.
A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arrangement is defined by a Weyl group we also provide an algebraic description, very handy for cohomology computations. In the last part we give a description in terms of tableaux for a toric arrangement appearing in robotics.
Given a surface F, we are interested in Z/2 valued invariants of immersions of F into R^3, which are constant on each connected component of the complement of the quadruple point discriminant in Imm(F,R^3). Such invariants will be called ``q-invariants.'' Given a regular homotopy class A in Imm(F,R^3), we denote by V_n(A) the space of all q-invariants on A of order <= n. We show that if F is orientable, then for each regular homotopy class A and each n, dim(V_n(A) / V_{n-1}(A)) <= 1.
First order invariants of generic immersions of manifolds of dimension nm-1 into manifolds of dimension n(m+1)-1, m,n>1 are constructed using the geometry of self-intersections. The range of one of these invariants is related to Bernoulli numbers. As by-products some geometrically defined invariants of regular homotopy are found.
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 …
This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connected symplectic 4-manifolds known to exhibit such property.
For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q+1. Similarly, in the case of odd q, the cobordism group of fol…
Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive homology class in E(1)_K when K is any nontrivial fibred knot in S^3. We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.
We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvature functional on the space of 2-sphere immersions into 3-space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8πand we establish a non-injectivity result for local minima.
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.
We use surgery along 2-tori embedded in a union of two copies of a product of punctured 2-tori to produce a new collection of homotopy 4-spheres (4-manifolds homotopy equivalent to $S^4$ and hence homeomorphic to $S^4$ but possibly not diffeomorphic to $S^4$). It is still unknown if these new examples are in fact exotic.
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.
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…
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…
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…
Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps …
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…
