Questions tagged [differential-topology]
Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
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Where and when are forms actually mandatory for differential geometry?
In the books that I've read (primarily Lee's Introduction to Smooth Manifolds, Tu's Introduction to Manifolds, and Arnol'd's Mathematical Methods of Classical Mechanics) I've seen cases where the ...
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Questions / Understanding Check on the Curvature of the Chern Connection.
Note: The background for this question / the understanding part of this question got very long. Sorry about that! It might suffice to skip down to (My main question) below.
I'm looking into positivity ...
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Canonical Identification of Tangent Vector of Matrix Lie Group
Question Setup
(See edits made for comments below)
In an abstract sense, one of the defintions that we can use to define a tangent vector at a point $p\in M$, where $M$ is a differential manifold $(M,\...
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Sanity check for a proof conserning outwards pointing normal vector on the boundary of a closed tubular neighborhood.
From Madsen and Tornehave book ''From calculus to cohomology'' in the proof for theorem 11.27.
Let $M$ an embedded (smooth) closed submanifold $M^n\subset \mathbb R^{n+k}$ and a vector field $X$ on $M$...
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Homotopy Extension Theorem: Is 'Retract of an Open Subset' a Weaker Condition?
${1.4. \textbf{Theorem}}$
Let $Z$ be a closed subspace of a normal space $N$. Let $f: N \to P$ be a continuous map, and let $g: Z \times I \to P$ be a homotopy of $f|Z$. If $g$ extends to a homotopy ...
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Involution map on real stiefel manifold orientation preserving?
We have an involution map on real Stiefel manifold $V_{n,2}$. $(v,w) \rightarrow (-v,-w)$. I want to know if this map is orientation preserving. n is odd here. My guess is, it is. As, Projective ...
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Whitney Embedding Theorem - construction of a specific atlas
In Hirsch's book, in order to prove Whitney Embedding Theorem it is first proven that every compact manifold $M$ can be embedded in some Euclidean space. To this end, Hirsch claims that one can easily ...
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Approximation Theorem of Whitney embedding
I was reading page 26 of Morris W. Hirsch's Differential Topology and came across the following approximation theorem. Could you tell me where I can find a more detailed and updated discussion of this ...
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Index of a nowhere vanishing vector field.
Let $M$ be a closed (smooth) manifold. For a (smooth) vector field $X$ on $M$ with only isolated zeros $\{p_1,\dots,p_k\}=X^{-1}\{0\}$, we define the $\text{Index}(X)=\sum_{p\in X^{-1}\{0\}}\iota(V,p)$...
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Can we define differentiable structures on topological spaces which are not manifolds?
I've been thinking about exactly how one determines differentiability of continuous maps between topological spaces. If $f$ is a map between topological vector spaces, the general idea is that $f$ ...
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How to understand that a $2$-twisted Möbius bundle is trivial and isomorphic to $\mathbb{S}^1 \times \mathbb{R}$?
Define the “$2$-twisted Möbius bundle” as the rank-$1$ subbundle $D$ of $E = \mathbb{S}^1 \times \mathbb{R}^2$ defined:
$D_{\mathrm{e}^{\theta \mathrm{i}}} = \operatorname{span}\{\cos(\theta) e_1 + \...
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If the boundary of a set is a $C^1$ manifold must its epsilon neighborhood also have a boundary of $C^1$ manifold?
Let $K\subset\mathcal{R}^n$ be compact and suppose it has a $C^1$ boundary, then I think the $\epsilon$-neighborhood of $K$ must have some form of smoothness for its boundary for any $\epsilon >0$. ...
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Mod 2 Intersection Number of $f\times f$
Let be $n,m\in \mathbb{N}$ such $m=2n$ and $n\geq 1$, and $f:\mathbb{S}^{n}\to \mathbb{S}^{m}$ a differentiable function.
Show that $I_{2}(f,f)=0$.
Here:
$$\triangle_{~\mathbb{S}^{m}}:=\{(s,s)\mid s\...
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References for Intersection Forms
I am currently learning the background materials towards a project on Seiberg-Witten invariants and 4-manifolds. Currently, I am trying to understand the ideas of intersection forms on 4-manifolds and ...
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Parallelizability and homotopy type
Suppose $M$ is a smooth, compact manifold with nonempty boundary, and that $\operatorname{int}(M)=M-\partial M$ is a parallelizable manifold. Then is it true that $M$ is also parallelizable? Actually ...
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Confusion regarding Adams spectral sequence and unoriented cobordism
So this probably the simplest (and thus trivial) place to look at the Adams spectral sequence but I thought it would be good for understanding since I know what the unoriented cobordism ring should ...
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The Homotopy Type of $Emb(\mathbb{R}^n, \mathbb{R}^n)$
I have encountered two statements respect to the homotopy type of $Emb(\mathbb{R}^n, \mathbb{R}^n)$.
One is that it is homotopically equivalent to $O(n)$ via Gram Schmidt Process. Another is that it ...
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Hirsch's proof of Morse–Sard Theorem
I am studying Morse–Sard Theorem from Hirsch's book Differential Topology and I fail to understand the proof of one of the preparatory lemmas.
Proposition. Let $M$ and $N$ be manifolds with $\dim M &...
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Universal cover of a Riemannian surface
Consider an open surface $\Sigma$, we embed it in $R^3$ and assume this embedding is proper so its image is an embedded Riemannian surface (the metric is induced from $R^3$).
Now we look the universal ...
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What is the comapctness argument mentioned here?
Following is the proof of Darboux's Theorem in John M Lee's Book. Can someone explain the compactness argument mentioned in the proof? I have highlighted the part using the compactness argument in the ...
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A smooth map of a sphere to itself is homotopic to a map with isolated fixed points
Let $v:S^k\to S^k$ be a smooth map of a sphere into itself. Such a map possibly can have nonisolated fixed points (e.g. the identity map of $S^k$). Can we always homotope $v$ to a smooth map $S^k\to S^...
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can we inverse the definition of vector bundles and covector bundles
In differential manifolds, vectors are defined as derivations of functions on manifold $M$, typically $v=\sum_i v_i \frac{\partial}{\partial x_i}$, and covectors are defined as linear functions of ...
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Double Tangent Bundle and Tangent and Normal Components to Acceleration
Consider a standard 4-bar mechanism (say a Grashof crank-rocker) with mobility $df = 1$ and its 1D configuration space (c-space) $Q \approx S^1$.
The c-space is essentially $\theta_2$, since, using ...
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Smoothness of a Function from a Manifold to $\mathbb{R}^{+}$
Let be $X\subseteq \mathbb{R}^{N}$ a diferentiable manifold.
Show that, if there is $U$ an open set in $\mathbb{R}^{N}$ such $X\subseteq U$, then there exists a differentiable function $\varepsilon:X\...
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Fubini-Study form is invariant under the action of $U(n+1)$
I am doing an exercise about the Fubini-Study form but i have some problem to prove the following fact : The Fubini-Study form $\omega$ is invariant under the standard action of $U(n+1)$ on $\mathbb{...
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Approximation and homotopy
In Differential Topology the usual strategy to discuss homotopy classes by differential means is approximation, because a manifold $M$ is a euclidean neigborhood retract (ENR) and any good ...
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Path Connectivity of the Space of Long Knots
I am reading about the space of long knots from the book Cubical Homotopy Theory by Munson and Volic (link).
Let $\iota \colon \mathbb{R} \hookrightarrow \mathbb{R}^d$ be the linear embedding defined ...
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Can the manifold boundary equation $\partial W=M$ possess more than 1 contractible solution?
Are there two non homeomorphic compact smooth contractible manifolds $W_1,W_2$ such that their boundaries are diffeomorphic manifolds? Is there an example of this situation such that the boundary is ...
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separating surface in manifold
Suppose $M$ is an orientable 3-manifold, $\Sigma$ is a closed embedded orientable surface in $M$. Suppose the inclusion map induces a surjective map from $\pi_1(\Sigma)\rightarrow \pi_1(M)$. It seems ...
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Geodesic curve, why dervative of length functional variation is in tangent space?
I was looking at this pdf. In page3,
Don't understand why that underlined statment is correct. Although the pdf has a picture
I was thinking to show $\langle \frac{d}{d\lambda}|_0~\alpha_\lambda (s),...
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Does the manifold bounday equation $\partial W=M$ has at most 1 solution?
Are there two non homeomorphic compact smooth manifolds $W_1,W_2$ such that their boundaries are diffeomorphic manifolds? Is there an example of this situation such that the boundary is a ...
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A theorem about isotopy similar to Whitney embedding
Suppose $M$ is a smooth manifold and there are two embeddings $f_0,f_1:M\rightarrow\mathbb R ^{n}$. If $n\ge 2\dim M+3$, then $f_0,f_1$ are isotopic; moreover there is a embedding $F:M\times [0,1]\...
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Equivalent Definition of a Smooth Map Between Manifolds
I am reading John Lee's Intro to Smooth Manifolds at the moment. In the text, two definitions of a smooth map $F:M\to N$, where $M$ and $N$ are smooth manifolds, are given, and it is stated that they'...
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Does every regular foliation admits a transversal regular foliation?
Suppose $M$ is a smooth manifold (compact, orietnted, or some other extra nice conditions), let $D\subset TM$ be an integrable distribution (i.e, $[D,D]\subset D$), then by Frobenius theorem, $D$ ...
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Quotient space of $Z_2$ involution of $S_2 \times S_2$
I need to find a $Z_2$ involution of $S^2 \times S^2$ that has the same isometries as $S^2 \times S^2$ and has $S^1 \times S^1$ as fixed points. From https://mathoverflow.net/questions/20836/...
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Trivialization of the tangent bundle of sphere
Can someone give an explicit trivialization of the tangent bundle of the sphere?
I know that the tangent bundle $\pi:T\mathbb{S}^2\to \mathbb{S}^2$ of the sphere $\mathbb{S}^2\subset\mathbb{R}^3$ is ...
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Is there a resource on the differential topology of Banach manifolds?
are there any resources that study the topology of spaces $C^k(M,N)$, where $M$ and $N$ are Banach (maybe even Frechet?) manifolds? I am interested in this since a a sizeable chunk of the book Im ...
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Is $X\times\{0\}\cup X\times \{1\}$ diffeomorphic to $X$?
Given $X$ a $n$-differentiable manifold such $\partial X=\varnothing$, prove there exists $Y$ a $n+1$-differentiable manifold with frontier, such $\partial Y$ is diffeomorphic to $X$.
My attemp: Let ...
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Compatibility relations between atlases (differential manifolds) [duplicate]
I'm starting to study Antoni A. Kosinski, Differential Manifolds and have stalled at the second page.
A chart over a topological space $M$ is defined as an open subset $U$ of $M$ with a homeomorphism $...
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Smooth circle bundles over blowup of weighted proective space
Consider the toric variety $X$ given by the (possibly weighted) blowup of the weighted $\mathbb{CP}_{[a_0, a_1, a_2]}$ in two of the three fixed points. If I am not mistaken, the second cohomology of $...
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Most general definition of a geodesic including non-metric spaces and non-affine connections?
Influenced by General Relativity, I originally thought geodesics were only defined on metric spaces. But I realized later that the definition is more general and does not require a metrizable space. ...
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Orientation of 1-manifolds with boundary in Milnor's Topology from the Differentiable Viewpoint
In the book, Milnor says that the orientation on each tangent space must be defined so that there is a neighborhood of each point and a chart $f$ from that neighborhood to $R^m$ or $H^m={(x_1,...,x_n)\...
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Given that $A\subset S$ are both regular surfaces, show that $A$ is open in the topology of $S$.
Exercise 14 of section 2-4 of (the Dover published translation of) do Carmo's "Differential Geometry of Curves and Surfaces" asks to show that a subset $A\subset S$ of a regular surface $S$ ...
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Intersection of product of cycles with graph equals intersection of pushforward
I'm trying to work out the proof of Lemma 2.5 in these notes: fix $A, B \in H_\bullet (X)$ of degrees adding up to $n$, the dimension of $X$. Let $f : X \to X$ be a continuous map, and let $\Gamma(f) \...
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Differential of map $\mathbb{R}^8\to\mathbb{R}^8$
I have this map $f:\Omega\subset\mathbb{R}^8\to\mathbb{R}^8$, defined by
$$
f(x_0,\dots,x_7)=(x_0^2-x_1^2-x_2^2-x_3^2,2x_0x_1,2x_0x_2,2x_0x_3,2x_0x_4,2x_0x_5,2x_0x_6,2x_0x_7)
$$
over the set $$\Omega=\...
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The bundle structure on $E/G = \bigcup_{x\in M} E_x/G_x$
I'm learning about the bundle structure on $E/G$ where $E$ is a vector bundle over $M$, and $G$ is a subbundle. I've reproduced my understanding below and asked some questions. Thanks for your help!
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$f$ has full rank $\implies$ $f$ is a covering map
Exercise 1.84 (d) of [GHL], it asks us to prove that
If two $n$-dimensional manifolds $\tilde M$ and $M$ are compact and if $M$ is connected, show that a smooth map $f : \tilde M\to M$ is a covering ...
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Restriction of vector bundle in case of non empty boundary
Let E and M be manifolds with boundary where E is a vector bundle over M. Consider the restriction of the vector bundle E to a vector bundle E_s over an embedded submanifold with boundary S of M. Is ...
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Do Smooth Quotient Maps on Manifolds Always Locally Resemble Projections?
I'm investigating the relationship between smooth submersions and smooth quotient maps in the context of smooth manifolds, and I'm particularly interested in understanding if every smooth quotient map ...
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Is every strong deformation retract of a manifold a very strong deformation retract?
Given a strong deformation retract of smooth manifolds $N\subset M$, $p:M\rightarrow N$ together with a homotopy relative to $N$ that witnesses the strong deformation retract $H:M\times [0,1]\...
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