In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provide a generalization of a classical genus estimate due to V.A. Rokhlin to the case of immersed surfaces.
We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.
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.
Sz\H ucs proved in 2000 that the $r$-tuple-point manifold of a generic immersion is cobordant to the $Σ^{1_{r-1}}$-point manifold of its generic projection. Here we slightly extend this by showing that the natural mappings of these manifolds are bordant to each other. The main novelty of our approach is that we construct the bordism explicitly.
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…
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.
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.
The cobordism group $N(M^n)$ of codimension-one immersions in the $n$-manifold $M^n$ has a natural filtration induced by any cellular decomposition. The problem addressed in this paper is the explicit computation of the graded group $gr^*N(M^n)$. We introduce some new invariants for immersions enlightening the Atiyah-Hirzebruch spectral sequence associated to $N(M)$, which are of combinatorial-geometric nature. Explicit computations are developed for $n\leq7$, and the group structure is also investigated for orientable 4-manifolds.
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.
A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3--manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these immersions. This is used to develop and implement an algorithm to compute the unit ball of the Thurston norm for cusped hyperbolic 3--manifolds of finite volume. As an application of independent interest, we give an upper bound on the minimal entropy of pseudo-Anosov maps of surfaces with number of cusps bounded linearly in genus.
This paper is to study the $\mathbb{R}^{3}$ case of \hyperref[Zhao]{[9]}. We determine all equivalence classes of immersed $3$-manifolds bounded by an arbitrary immersed surface in $\mathbb{R}^{3}$.
Given any finite subset X of the sphere S^n, n>1, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space R^{n+1} whose Gauss map misses X. In particular, this answers a question of M. Gromov.
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.
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 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 study minimal immersions of closed surfaces (of genus $g \ge 2$) in hyperbolic 3-manifolds, with prescribed data $(σ, tα)$, where $σ$ is a conformal structure on a topological surface $S$, and $αdz^2$ is a holomorphic quadratic differential on the surface $(S,σ)$. We show that, for each $t \in (0,τ_0)$ for some $τ_0 > 0$, depending only on $(σ, α)$, there are at least two minimal immersions of closed surface of prescribed second fundamental form $Re(tα)$ in the conformal structure $σ$. Moreover, for $t$ sufficiently large, there exists no such minimal immersion. Asymptotically, as $t \to 0$…
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 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.
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…
Immersions of punctured 4-manifolds: with applications to Quantum Cellular Automata
Microsoft Research, Station Q, and Department of Mathematics University of California Santa Barbara（微软研究院、Station Q 和加州大学圣塔芭芭拉分校数学系）；School of Mathematical Sciences University of Southampton（南安普顿大学数学科学学院）；Hausdorff Center for Mathematics（豪斯多夫数学中心）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）
We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh. Nonetheless, in the setting of geometrically finite groups whose limit sets are circle packings, we derive 2 density criteria for non-closed geodesic plane immersions, and show that closed immersions give rise to surfaces with finitely generated fundamental groups. We also obtai…
The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism group of codimension one immersions in the manifold. The connected sums of manifolds and immersions induce multiplicative structures which are respected by this surjection. We prove that those cubulations which map combinatorially into the standard decomposition of ${\bf R}^n$ for l…
As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds and is given by the non-vanishing of the $α$-genus of Hitchin. When unobstructed we shall realize a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure for constructing positive scalar curvatur…
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…
In this paper we investigate the relationships between closed AdS 3-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for the volumes. We also find applications to the theory of minimal immersions into quadrics with their natural pseudo-Riemannian structure: using the geometry of the AdS manifolds we can characterize the representations admitting equivariant minimal immersions of the Poincare disc into the Klein quadric, the Grassmannian Gr(2,4), an…
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.
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.
This thesis mainly treats two developments of the classical theory of hypersurfaces inside pseudo-Riemannian space forms. The former - a joint work with Francesco Bonsante - consists in the study of immersions of smooth manifolds into holomorphic Riemannian space forms of constant curvature -1 (including SL(2,C) with a multiple of its Killing form): this leads to a Gauss-Codazzi theorem, it suggests an approach to holomorphic transitioning of immersions into pseudo-Riemannian space forms, a trick to construct holomorphic maps into the PSL(2,C)-character variety, and leads to a restatement of B…
Let $f:V^n\looparrowright M^m$ be a smooth generic immersion. Then the set of points, that have at least $k$ preimages is an image of a (non-generic) immersion. If the manifolds $V^n$ and $M^m$ are oriented and $m-n$ is even, then the manifold of $k$-fold points is also oriented. In this paper we compute the oriented bordism class of the manifold of $k$-fold points in terms of the differential $df$, provided the tangent bundle of the manifold $M^m$ has a nowhere zero cross-section.
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.
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 establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding $S^1\times S^2\to\R^6_\st$ with vanishing Maslov class.
