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 $F_8 : T^3 \hookrightarrow {\bf R}^5$ of the 3-torus $T^3$ with the following properties: (1) $F_0 \sharp h$ is regularly homotopic to $F_8$ for some immersion $h : S^3 \looparrowright {\bf R}^5$, and (2) the immersion $h$ as above cannot be chosen from a regular homotopy class containing an embedding.
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.
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.
We use the wrinkling theorem proven in Y. Eliashberg and N. Mishachev, "Wrinkling of smooth mappings and its applications - I", Invent. Math., 130(1997), 345-369, to fully describe the homotopy type of the space of S-immersions, i.e. equidimensional folded maps with prescribed folds.
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.
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.
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.
We give three formulas expressing the Smale invariant of an immersion f of a (4k-1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k-1)-space. The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k-1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k-1)-space. Moreover, any generic homotopy in (6k-1)-space connecting f to g must have at lea…
In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group of spheres $Π_n$ there is no elements with Hopf invariant one. The new proof is based on geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed by R.Wells) about the representation of stable homotopy groups of the real projective infinite-dimensional space (…
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.
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…
Given smooth manifolds $V^n$ and $M^m$, an integer $k$, and an immersion $f:V\looparrowright M$, we have constructed an obstruction for existence of regular homotopy of $f$ to an immersion $f':V\looparrowright M$ without $k$-fold points. This obstruction takes values in certain framed bordism group, and for $(k+1)(n+1)\leq km$ turns out to be complete.
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 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.
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.
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…
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 …
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…
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.
We show that if a compact, oriented 4-manifold admits a coassociative-free immersion into the Euclidean 7-space then its Euler characteristic and signature vanish. Moreover, in the spin case the Gauss map is contractible, so that the immersed manifold is parallelizable. The proof makes use of homotopy theory in particular obstruction theory. As a further application we prove a non-existence result for some infinite families of 4-manifolds that can not be addressed previously. We give concrete examples of parallelizable 4-manifolds with complicated non-simply-connected topology.
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 …
Two locally generic maps f,g : M^n --> R^{2n-1} are regularly homotopic if they lie in the same path-component of the space of locally generic maps. Our main result is that if n is not 3 and M^n is a closed n-manifold then the regular homotopy class of every locally generic map f : M^n --> R^{2n-1} is completely determined by the number of its singular points provided that f is singular (i.e., f is not an immersion).
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…
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.
By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a regular homotopy version of the Turaev cobracket.
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…
Let N and P be smooth manifolds of dimensions n and p (n \geq p \geq 2) respectively. Let Ω(N,P) denote an open subspace of J^{infty}(N,P) which consists of all regular jets and jets with prescribed singularities of types A_{i}, D_{j} and E_{k}. An Ω-regular map f:N \to P refers to a smooth map having only singularities in Ω(N,P) and satisfy the transversality condition. We will prove the homotopy principle in the existence level for Ω-regular maps.
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.
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 …
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 …
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.
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…
Let $ M^{n+1} $ ($ n \ge 2 $) be a simply-connected space form of sectional curvature $ -κ^2 $ for some $ κ\geq 0 $, and $ I $ an interval not containing $ [-κ,κ] $ in its interior. It is known that the domain of a closed immersed hypersurface of $ M $ whose principal curvatures lie in $ I $ must be diffeomorphic to the sphere $ S^n $. These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed $ I $, the space $ \mathscr{F} $ of all such closed hypersurfaces is either empty or weakly homotopy equivale…
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 text is based on my talk at the popular science conference ``Dark geometry fest'' which was related to geometric methods and their applications, July 17, 2022. We will move towards the Smale-Hirsch theorem. To this end we will deal with the simplest partial case of this theorem -- the Whitney-Graustein theorem on regular curves in the plane. The text is written in an accessible and sometimes informal way, it is intended primarily for people who are interested in mathematics, but have not yet studied it deeply enough.
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 prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with respect to a compatible symplectic form.
The abstract link L_d of the complex isolated singularity x^2 + y^2 + z^2 + v^{2d} = 0 is diffeomorphic to S^3 \times S^2. We classify the embedded links of these singularities up to regular homotopies precomposed with diffeomorphisms of S^3 \times S^2. Let us denote by i_d the inclusion of L_d in S^7. We show that for arbitrary diffeomorphisms φ_d of S^3 \times S^2 with L_d the compositions i_d \circ φ_d are image regularly homotopic for two values d_1 and d_2 if and only if d_1-d_2 is even.
Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K(Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh(*) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S^1-transfer map t: SigmaCP^infty--> S. The homotopy groups of Wh(*) are determined in a range of degrees, and the cohomology of Wh(*) is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interp…
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 investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regular CW-complexes, which are a sort of combinatorial-up-to-homotopy objects, are modeled (up to homotopy) by their associated finite spaces. This is accomplished by generalizing a classical result of McCord on simplicial complexes.
In this paper we compute the singular homology of the space of immersions of the circle into the $n$-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together with loop products, this offers some new computational tools for string topology.
We show that the hyperbolization of polyhedra pulls back regular neighborhoods of PL submanifolds. Applying this to the Riemannian version of the hyperbolization due to Ontaneda gives open complete manifolds of pinched negative curvature that are homotopy equivalent to closed smooth manifolds but contain no smooth spines. We also find open complete negatively pinched manifolds that are homotopy equivalent to closed non-smoothable manifolds.
