Embeddings from the point of view of immersion theory: Part I

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.

Embedding Calculus, as described by Weiss, is a calculus of functors, suitable for studying contravariant functors from the poset of open subsets of a smooth manifold M, denoted O(M), to a category of topological spaces (of which the functor Emb(-,N) for some fixed manifold N is a prime example). Polynomial functors of degree k can be characterized by their restriction to O_k(M), the full subposet of O(M) consisting of open sets which are a disjoint union of at most k compone...

Let $M$ be a simply-connected $m$ dimensional manifold of finite type and $k$ a positif integer. In this paper we show that the rational Betti numbers of each component of the space of immersions of $M$ in $\mathbb{R}^{m+k}$, have polynomial growth. As consequence, we deduce that, if $M$ is a manifold with Euler characteristic $\chi\left(M\right)\leq -2$, the Betti numbers of smooth embeddings, $Emb\left(M ,\mathbb{R}^{m+k}\right)$, have exponential growth if $k\geq m+1$. The...

Let $N$ be a closed orientable connected $n$-manifold, $n\ge 4$. We classify embeddings of the punctured manifold $N_0$ into $\R^{2n-1}$ up to isotopy. Our result in some sense extends results of J.C. Becker -- H.H. Glover (1971) and O. Saeki (1999).

In this paper we study the rational homotopy of the space of immersions, $Imm\left(M,N\right)$, of a manifold $M$ of dimension $m\geq 0$ into a manifold $N$ of dimension $m+k$, with $k\geq 2$. In the special case when $N=\mathbb{R}^{m+k}$ and $k$ is odd we prove that each connected component of $Imm\left(M,\mathbb{R}^{m+k}\right)$ has the rational homotopy type of product of Eilenberg Mac Lane space. We give an explicit description of each connected component and prove that i...

Aim of this note is to extract cohomological information about the manifold $Emb(M,N)$ from the topology of the target manifold N. For special conditions, a monomorphism $H^1 (N) \to H^1 (Emb(M,N))$ is constructed.

Let $Z$ be a smooth compact $(n+1)$-manifold. We study smooth embeddings and immersions $\beta: M \to Z$ of compact or closed $n$-manifolds $M$ such that the normal line bundle $\nu^\beta$ is trivialized. For a fixed $Z$, we introduce an equivalence relation between such $\beta$'s; it is a crossover between pseudo-isotopies and bordisms. We call this equivalence relation ``{\sf quasitopy}". It comes in two flavors: $\mathsf{IMM}(Z)$ and $\mathsf{EMB}(Z)$, based on immersions ...

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 mathem...

We prove convergence of Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two, so in particular for diffeomorphisms between surfaces. We also relate the Johnson filtration of the mapping class group of a surface to a certain filtration arising from embedding calculus.

It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is Z_2. If the degree of the class is smaller or equal to 6 or equal to k-2 or k-1 under the condition that the coefficient ring is the integer rin...