我们的工作是平稳的。设N是一个闭连通n-流形，且m>n+2。用E^m（N）表示直到合痕的嵌入N -> R^m的集合。群E^m（S^n）通过流形与球面的嵌入连通和作用在E^m（N）上。如果E^m（S^n）不为零（这经常发生在2 m <3 n +4时），那么这个作用没有结果，也没有E^m（N）的完整描述。我们的主要结果是这个作用的平凡性和有效性的例子，以及对某些4-流形N嵌入到R^7中的完全合痕分类。证明是基于手术理论的Kreck修改和一个新的嵌入不变量的建设。推论。(a)在等位上存在唯一的嵌入CP^2 -> R^7。(b)对于每个嵌入f：CP^2 -> R^7和每个非平凡结g：S^4 -> R^7，嵌入f#g与f是同位素的。
英文摘要：
We work in the smooth category. Let N be a closed connected n-manifold and assume that m>n+2. Denote by E^m(N) the set of embeddings N -> R^m up to isotopy. The group E^m(S^n) acts on E^m(N) by embedded connected sum of a manifold and a sphere. If E^m(S^n) is non-zero (which often happens for 2m<3n+4) then no results on this action and no complete description of E^m(N) were known. Our main results are examples of the triviality and the effectiveness of this action, and a complete isotopy classification of embeddings into R^7 for certain 4-manifolds N. The proofs are based on the Kreck modification of surgery theory and on construction of a new embedding invariant.
Corollary. (a) There is a unique embedding CP^2 -> R^7 up to isoposition.
(b) For each embedding f : CP^2 -> R^7 and each non-trivial knot g : S^4 -> R^7 the embedding f#g is isotopic to f.
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.
In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.
We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary. This embedding obstruction can also be used to detect exotic smooth structures on 4…
In this note, we discuss embeddings of $3$--manifolds via open books. First we show that every open book of every closed orientable $3$--manifold admits an open book embedding in any open book decompistion of $S^2 \times S^3$ and $S^2 \widetilde{\times} S^3$ with the page a disk bundle over $S^2$ and monodromy the identity. We then use open book embeddings to reprove that every closed orientable $3$--manifold embeds in $S^5.$
We discuss embedding of manifolds in the category of open books, contact manifolds and contact open books. We prove an open book version of the Haefliger--Hirsch embedding theorem by showing that every $k$-connected closed $n$-manifold ($n\geq 7$, $k < \frac{n-4}{2}$) admits an open book embedding in the trivial open book of $\mathbb{S}^{2n-k}$. We then prove that every closed manifold $M^{2n+1}$ that bounds an achiral Lefschetz fibration, admits open book embedding in the trivial open book of $\mathbb{S}^{2\lfloor\frac{3n}{2}\rfloor + 3}$. We also prove that every closed manifold $M^{2n+1}$ b…
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Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens space smoothly embeds in n copies of the complex projective plane.
In this note we observe that one can contact embed all contact 3-manifolds into a Stein fillable contact structure on the twisted $S^3$-bundle over $S^2$ and also into a unique overtwisted contact structure on $S^3\times S^2$. These results are proven using "spun embeddings" and Lefschetz fibrations.
Spherical alterations of handles: embedding the manifold plus construction
在我们早期关于流形端点的工作中，一个关键的工具是能够将Quillen Plus Construction提供的协边嵌入到这些端点中。在这里，我们开发了一个“球形修改”的技巧，提供了一个建设性的方法来获得这样的嵌入。更重要的是，这种方法允许更一般的嵌入结果。在本文中，我们开发的推广，加上建设和相应的协边如何可以嵌入到流形满足适当的基本群性质。这里得到的结果在我们正在进行的非紧流形的研究中起着重要的作用。
We consider embeddings of 3-manifolds in $S^4$ such that each of the two complementary regions has an abelian fundamental group. In particular, we show that an homology handle $M$ has such an embedding if and only if $π_1(M)'$ is perfect, and that the embedding is then essentially unique.
This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov--Pyatetski-Shapiro and Agol--Belolipetsky--Thomson non-arithmetic manifolds embed geodesically. Moreover, we show that the number of commensurability classes of hyperbolic manifolds with a representative of volume $\leq v$ that bounds geometrically is at least $v^{Cv}$, for $v$ large enough.
We prove that for any compact orientable connected 3-manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open 2-disk removed admits a Lagrangian embedding into the standard symplectic 6-space. Moreover, minimal Maslov number of the Lagrangian embedding is equal to 1.
We show that, for $\varepsilon=\dfrac{1}{4000}$, any action of a finite cyclic group by $(1+\varepsilon)$-bilipschitz homeomorphisms on a closed 3-manifold is conjugated to a smooth action.
Suppose a finite group $G$ acts on a manifold $M$. By a theorem of Mostow, also Palais, there is a $G$-equivariant embedding of $M$ into the $m$-dimensional Euclidean space $\RR^{m}$ for some $m$. We are interested in some explicit bounds of such $m$. First we provide an upper bound: there exists a $G$-equivariant embedding of $M$ into $\RR^{d|G|+1}$, where $|G|$ is the order of $G$ and $M$ embeds into $\RR^d$. Next we provide a lower bound for finite cyclic group action $G$: If there are $l$ points having pairwise co-prime lengths of $G$-orbits greater than $1$ and there is a $G$-equivariant …
We prove that any arithmetic hyperbolic $n$-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold or its universal $\mathrm{mod}~2$ Abelian cover can.
We consider embeddings of 3-manifolds $M$ in $S^4$ such that the two complementary regions $X$ and $Y$ each have nilpotent fundamental group. If $β=β_1(M)$ is odd then these groups are abelian and $β\leq3$. In general, $π_1(X)$ and $π_1(Y)$ have 3-generator presentations, and $β\leq6$. We determine all such nilpotent groups which are torsion-free and have Hirsch length $\leq5$.
We exhibit an infinite family of rational homology balls which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson's diagonalisation theorem, and use this to show that no two of our examples may be embedded disjointly.
The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, ξ_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, ξ_N),$ provided $M$ contact embeds in $(N, ξ_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $ξ_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sp…
Whenever a finitely generated group $G$ acts properly discontinuously by isometries on a metric space $X$, there is an induced uniform embedding (a Lipschitz and uniformly proper map) $ρ: G \rightarrow X$ given by mapping $G$ to an orbit. We study when there is a difference between a finitely generated group $G$ acting properly on a contractible $n$-manifold and uniformly embedding into a contractible $n$-manifold. For example, Kapovich and Kleiner showed that there are torsion-free hyperbolic groups that uniformly embed into a contractible $3$-manifold but only virtually act on a contractible…
