Questions tagged [differential-topology]

The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.

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34 views

Local coordinates of one form on a principal bundle

I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates. Let's say ...
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Restrictions on pointed lifts of isometries

Let $M$ be a closed Riemannian manifold and let $f$ be an isometry of $M$ that fixes a point $\ast \in M$ and acts trivially on $\Gamma := \pi_1(M,\ast)$. Then there is a unique isometry $\tilde{f}$ ...
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Extending an embedding with trivial normal bundle

I am recently studying the book Notes on Cobordism Theory by R. E. Stong and I have noticed that the proposition below is (implicitly) used (for example to extend a $(B,f)$ structure on a boundary of ...
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Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?

This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question. Suppose we have two three-...
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Diffeomorphism classification of manifolds with fundamental group $\mathbb{Z}_n$, $n>2$

I am looking for diffeomorphism classifications of manifolds with $\pi_1=\mathbb{Z}_n$, $n>2$. I only know of Ottenburger's PhD thesis, A diffeomorphism classification of 5- and 7-dimensional non-...
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What locales correspond to Manifolds?

I am studying the categorical equivalence between (sober) topological spaces and (spatial) locales with enough points. As the title implies, I am interested in finding localic analogues of both ...
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Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence

Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to any self-homotopy ...
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61 views

Uniqueness of collar neighborhoods for non-compact boundary case in smooth setting

Let $M$ be a smooth manifold and let $f_0, f_1 \colon [0, 1] \times \partial M \to M$ be two smooth embeddings that are the identity map on $\partial M \times\{0\} = \partial M$ . If $\partial M$ is ...
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Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?

I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
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271 views

Non orientable, closed manifold covered by two contractible charts

This is a follow up of my previous MO question "Non orientable, closed manifold covered by two simply-connected charts." Nick L's nice answer shows that such manifolds actually exist, ...
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Questions about a paper by Laudenbach and Poénaru

I am working on the 1972 paper A Note on 4-Dimensional Handlebodies by F. Laudenbach and V. Poénaru, and I had two questions. I will use their notations to simplify things, since the paper is very ...
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Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)

It seems that there is no digital copy of Leon Karp's Ph.D. thesis L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976. on internet and his paper excerpted from his thesis is very brief ...
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Reference request: Milnor rank of spheres

Milnor defines the rank of a smooth manifold $M$ as the maximum cardinality of a linearly independent set of vector fields on $M$ whose elements are pair wise commuting. In other words, the rank of $M$...
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226 views

Existence of a certain foliation of $\mathbb R^n$

Notation: We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse. Question: Let $n \geq 2$. Given a countable dense set of points $P \...
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The topology of spaces of immersed circles in the plane

Whitney number (the turning number of the tangent) classifies the connected components of the space of immersed circles in the plane. What is known about the topology of the connected components?
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Existence of developable ribbonization of a surface

Let $S$ be a smooth compact surface embedded in $\mathbb{R}^{3}$. It is well-known that there exists a triangulation of $S$. I am considering an alternative way of approximating $S$, where instead of ...
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Looking for a criterion for a subset of a complex variety to be of measure zero

Suppose $f:X\longrightarrow Y$ is a surjective morphism of smooth complex varieties. Let $S$ be a subset of $X(\mathbb{C})$. I'm wondering if there are results that roughly say that if the ...
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Dehn surgery along primitive knot in 3-dimensional handlebody

I'm studying the article "An alternative proof of Lickorish–Wallace theorem" (doi link) and I got stuck in a problem. Let $H_g$ be a 3 dimensional handlebody of genus $g$, a primate curve in ...
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Is each Lipschitz action of a finite group on the 3-sphere equivalent to a linear action?

It is known that each action of a compact group on the 2-dimensional $S^2$ sphere is equivalent (=conjugated) to the linear action of a subgroup of $O(3)$ on $S^2$. On the other hand, there exists a ...
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Godbillon–Vey invariant and leaf space of foliations

I recently got to know about the existence of the so-called Godbillon–Vey invariant, and I am interested in its relationship with foliation theory in 3-manifolds. I briefly recall here the definition: ...
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108 views

Collar neighborhood theorem for manifold with corners

I was reading this wonderful sequence of posts: nlab: manifold with boundary and nlab: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood ...
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Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group using De Rham cohomology

Cross-post from MSE. Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b_p\geq 1$. ...
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Classification of square roots of line bundles and metalinear/metaplectic structures

Reading some books and articles about geometric quantization I got confused about the classification of square roots of complex line bundles over a manifold. Consider the group of isomorphism classes ...
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Density of $G$-invariant morse functions

Let $G$ be a finite group acting on a compact manifold $M$. Let $f$ be a $G$-invariant smooth function. Can it be approximated by $G$-invariant Morse functions?
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Local topology of Whitney stratified spaces

Let $M$ be a smooth manifold, let $\mathcal{P}$ be a Whitney stratification of $M$ and let $S\subset M$ be a stratum with closure $\overline{S}$. Question: Does there exist an open neighborhood $U\...
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Antisymmetric tensor coordinates and tensorial spaces

I am currently working on some geometric aspects of higher-spin models for which there appear antisymmetric tensor coordinates $X^{\mu\nu}=-X^{\nu\mu}$, with $\mu,\nu=1,...,N$, which have been ...
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Fiberwise starshaped Hypersurfaces in $T^*M$

I just read the following assertion in this paper arXiv link : Let $\Sigma$ be a smooth connected Hypersurface in $T^*M$. We say that $\Sigma$ is fiberwise starshaped if for each point $q\in M$ the ...
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141 views

Subset of the domain of attraction

Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$ $$ \frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t)) $$ be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
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singular homology of manifold with corners

Given two smooth manifolds with corners, let's say that a map $f:X\to Y$ is "transversally smooth" if it is smooth in the usual sense and if (in a local sense on $X$) for every open Whitney ...
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1answer
107 views

Reference for non-parallel harmonic $k$-forms

I want to get some deep understanding on closed orientable Riemannian manifolds admitting $k$-forms ($k\geq 2$) $\omega$ that satisfices the following conditions: $$\nabla \omega\neq 0,\quad \Delta\...
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Equivariant smooth approximation

Suppose we have a compact manifold $M$ with the action of a compact group $G$. Consider the space of $C^l$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{l}(M)$ with the $C^l$ topology and the space ...
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Spin structures induced on embedded circles and choices of trivialisations

I have a presumably basic question concerning spin structures that has me a bit confused. Let $C$ be a circle embedded in a spun manifold $X^n$. Given a choice of trivialisation of the normal bundle ...
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Existence of closed transversals for taut foliations in arbitrary codimension

There are several different definitions of "tautness" for foliations, the most widely know is probably topological tautness, which is specific to codimension one and means that the foliation ...
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228 views

Is $U(n)$ a Kahler manifold?

I am wondering if it is known whether the unitary group $U(n)$ is a Kahler manifold, and, if so, what is a reference for this.
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555 views

Examples of Banach manifolds with function spaces as tangent spaces

I have recently been learning the theory of Banach manifolds through Serge Lang's book on Differential Manifolds. So far the objects seem rather interesting but my intuition always comes from the ...
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Fixed point set of diffeomorphism is a submanifold

I am in the following setting: Let $(\Sigma,\alpha\vert_\Sigma)$ be a compact regular energy surface of restricted contact type in an exact Hamiltonian manifold $(M,d\alpha,H)$. Given $\varphi \in \...
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restricting the “Whitney” map

$\newcommand\R{\mathbb R}$Suppose $f:\R^2 \to \R^2$ is a Whitney map with singularities (well, I'm not sure if this is the name for it, Whitney calls them excellent maps in his 1955 paper), i.e. it is ...
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Simple properties of the codifferential

The exterior derivative $d$ has many very nice algebraic relations. For example $d(\alpha\wedge\beta) = (d\alpha)\wedge \beta + (-1)^k\alpha\wedge(d\beta)$ $f^*(d \alpha)=d f^*(\alpha)$. $d\circ ...
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Almost isometric manifolds are diffeomorphic

I am looking for a reference to the following statement. (It should be known --- I saw it before, don't remember where; search by keywords did not help.) Let $f\colon M\to N$ be a homeomorphism ...
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Blowing up the zero section for “Chasse au Canard” (some new kind of geometric canards)

In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a ...
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Condition for existence of a continuous function realizing a partition

Let $\{U_i\}_{i=1}^{I}$ be a non-empty and finite collection of non-empty, disjoint, open, (and obviously bounded) subsets of $[0,1]^n$. Suppose also that $[0,1]^n=\cup_{i =1 }^{ I} \overline{U_i}$. ...
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1answer
102 views

Clarification of different notions of spin structures

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$I am confused about the equivalence of some various definitions of spin structures and I was hoping for some help clearing out the fog. Let ...
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107 views

Stokes's theorem for pseudo-differential forms

I recently came across the idea of a pseudo differential form (and "pseudo" objects in a manifold in general). Let $\Psi M$ is the real line bundle on $M$ which changes sign between two ...
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1answer
102 views

Approximating continuous functions via diffeomorphisms on compact manifolds

Let $M$ be a compact and connected manifold without boundary. My question is how to prove the following fact which I believe is true: If $f : M \to \mathbb{R}$ is a continuous function that attains ...
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Diffeomorphic excision of a manifold

Let $M$ be a connected differentiable manifold. How can one describe those closed subsets $A$ of $M$ such that there is a diffeomorphism $\varphi:( M - A )\to M$ ?
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Smooth dual cell structure

Let us consider a closed oriented smooth manifold M. It is well known that a smooth combinatorial triangulation can be constructed for it. That is to say, a homeomorphism from the geometric ...
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What is known about this conjectured symmetry in the generalized Radon-Hurwitz numbers?

The generalized Radon-Hurwitz number $\rho(m, n)$ is defined as the maximal dimension of a subspace contained in $Q_{m,n }$, the subset of all real $m\times n$ matrices $A$ which satisfy $AA^T=\lambda ...
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Elementary proof of an inequality for the Radon-Hurwitz numbers

Edit: In all likelihood, the original question does not have a positive answer (see comment by abx). Modified question: Let $\rho_H(n)$ be the maximal dimension of a space of symmetric real matrices ...
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Quotients of a fixed manifold by a fixed Lie group

Let $M$ a connected paracompact differentiable manifold. Let $G$ a connected Lie group. I am interested in the possible "regular" (e.g. smooth) quotients of $M$ by actions of $G$. What ...
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Do smooth maps with nowhere-maximal rank have small image?

I’m trying to better understand the concept of “maps with small image” as used by Lipyanskiy in his construction of “geometric homology” in https://arxiv.org/abs/1409.1121. Lipyanskiy utilizes ...

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