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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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Quotient space form by the action of the discrete Heisenberg group on the Heisenberg group

Though I am a beginner to differential topology, pardon me for something very basic. Here is my attempt! H(The set of $3 \times 3$ unipotent matrices over $\mathbb{R}$, Heisenberg group) is ...
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How to check when two manifolds $X_a$ and $Y$ intersect transversally

The question is to find the values of $a$ for which $X_a = \{(x,y,z)\in R^3 | x^2+y^2+z^2 = a\}$ and $Y=\{(x,y,z)\in R^3 | x+y^2+2z = 1\}$ intersect transversally in $R^3$. If $Y$ had $x^2$ instead ...
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Finding critical points of a function

$X = \{(x,y,z\in R^3 | x^2+2y^2-3z^2 = 1)\}$. I have proved earlier that X is a 2-manifold and that the tangent space $T_{(x,y,z)}(X) = \{(a,b,c)\in R^3 | 2xa+4yb-6zc = 0\}$. I have two parts to my ...
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Prove that $f$ between certain manifolds has a critical point

$X$ is a nonempty compact manifold and $Y$ is a connected non-compact manifold, and $f: X\rightarrow Y$ is a smooth function. Need to prove that $f$ has a critical point. My attempt: Let $f$ does not ...
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Does a bijective submersion imply diffeomorphism

Submersion, by definition is, $df_x: T_x(X)\rightarrow T_y(Y)$ is surjective and if it is also bijective, then intuitively it is a diffeomorphism for me as $df_x^{-1}:T_y(Y)\rightarrow T_x(X)$ is well ...
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Example in the proof of Sard Theorem

In the book Topology from the differential viewpoint - Milnor, there is a proof of Sard Theorem : When $f:X^{n+m}\rightarrow Y^n$ is a smooth map where $X^N$ has $N$ dimension, then $f(C^1)$ has a ...
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Cap product and de Rham cohomology

Let $M$ be a compact smooth $d$-dimensional oriented manifold. The natural pairing of $d$-forms $\omega^{(d)}$ with the fundamental class is given by integration $\int_M \omega^{(d)}$. Let us also ...
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Why is the intrinsic curvature of a sphere expressed in terms of an extrinsic parameter, its radius? [on hold]

Is there even such a thing as an extrinsic curvature of a sphere. Why can't its surface simply be its circumference squared divided by π , and could this be proved without reference to a space in ...
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1answer
62 views

Why $TM $ is trivial in this case?

If I have a $X_1,...,X_n $ vectors fields and a basis of $T_pM $ for all $p \in M $, Why the tangent bundle $TM \cong M × \mathbb{R}^n $ ?
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Universal principal bundle of a subgroup $H \subseteq G$.

This was claimed in the referenced link: If $H \subseteq G$ is an admissible subgroup, i.e. $G \rightarrow G/H$ is a principal $H$ bundle. Let $EG \rightarrow BG$ be a model universal ...
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Interpolation between Rotations

I have to define a continuous function g: [0, 1] $\rightarrow$ such that g(0) = I and g(1) = R $\in$ SO(3). I know we can do this kind of interpolation using quaternions and slerp. But, the question ...
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Reference request for gauge theory in low dimensional topology

I've been studying 3 and 4 manifold topology and it seems to me that lots of very powerful invariants come from a mysterious place called "gauge theory". When I peer into this place, I am confronted ...
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Continuity of Projection operator from Tangent Bundle to Manifold

I've been struggling with this problem for several hours now and I would appreciate some assistance. This is the only part of the larger problem that I am stuck on. The Tangent Bundle $TM$ of a ...
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1answer
43 views

Holomorphic Morse functions

For a holomorphic $f:\mathbb{C}^n\rightarrow \mathbb{C}$ and $a=(a_1, \dots, a_n) \in \mathbb{C}^n$, let $f_a:\mathbb{C}^n\rightarrow \mathbb{C}$ be the function $$(z_1, \dots, z_n) \mapsto a_1z_1 + ...
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Showing that there is not a global diffeomorphism between unit quaternions and $\mathrm{SO}(3)$

I am new to differential geometry. I have the following question: Let $\mathbf{Q}$ denote the set of unit quaternions. I already have proved using the implicit function theorem that $\mathbf{Q}$ is a ...
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Proving that unit quaternions are a 3 Manifold

I am very new to topology, and I am having trouble on how to prove if something is a manifold or not. The question states that: Let Q donate the set of unit quaternions (a) Show that Q is a 3-...
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Showing the Kodaira Map is injective

I'm trying to prove the Kodaira embedding theorem via peaked sections, (exercise 7.10 in Székelyhidi's "An Introductionto Extremal Kahler Metrics"). My issue isn't to do with peaked sections, rather I'...
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Connecting the Fibres of a Lefschetz Fibration With Disconnected Fibres.

The following is an exercise set by Chris Wendl in his book Holomorphic Curves in Low Dimensions. I'm fairly new to the subject, and wasn't sure how to approach it, so any help with it would be ...
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Surjective functions from a $n$-dimensional hypercube to $\mathbb{R}^m$ when $n > m$

I had asked a similar question before. Functions from an $n$-dimensional hypercube to $\mathbb{R}^m$ when $n >m$. I am wondering if there are any surjective functions. Let $n$ and $m$ be ...
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Square Torus homeomorphism

If we have a square torus on R$^2$ defined by the equivalence relation $(a_1, b_1)$ ~ $(a_2, b_2)$ if and only if $a_1 - a_2$ and $b_1 - b_2$ are integers, how would you show that they are ...
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Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
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1answer
71 views

Proving different projective planes homeomorphic?

I am having major trouble showing that the version of the projective plane here (with a Mobius strip) is homeomorphic to the projective plane that is defined as the quotient of the sphere $S^2$ via ...
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Euler’s Characteristic for nonorientable surfaces?

I recently learned about this in a topology class but wanted to know how to apply the Euler Characteristic for surfaces such as projective planes and Klein bottles (nonorientable surfaces). I have ...
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Functions from an $n$-dimensional hypercube to $\mathbb{R}^m$ when $n >m$.

Let $n$ and $m$ be integers such that $n > m$. Suppose there exists a $n$-dimensional hypercube in $\mathbb{R}^n$. Let the hypercube be divided into $2^n$ regions ($n$-dimensional volumes) by ...
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1answer
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Visualizing tangent space from its definition

Let me quote Guillemin, Pollack here. "We can use derivatives to identify the linear space that best approximates a manifold $X$ at a point $x$. Suppose $X\subset R^n, \phi:U\rightarrow X$ is a local ...
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A differentiable manifold class $C^k$ but not class $C^{k+1}$

Please show that the graph of $f(x)=|x|^λ$, where $k<λ<k+1$ is a differentiable manifold of class $C^k$ but not class $C^{k+1}$. (Problem in Differential Topology, Hirsch, Morris W.) Or, please ...
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Spaces of submanifolds

Let $M$ and $N$ be smooth manifolds with $\dim M<\dim N$. The spaces $\mathrm{Emb}(M,N)\subset\mathrm{Imm}(M,N)$ of smooth embeddings and immersions $f:M\to N$, respectively, are infinite ...
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Does a surface $S$ diffeomorphic to a sphere have an umbilical point?

I should see if a surface which is diffeomorphic to a sphere in $\mathbb{R}^3$ always has an umbilical point. So, I start by locally parametrizing the sphere with a diffeomorphism $x: \Omega \to S^2$ ...
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Why does any connected closed $m$-manifold that can be embedded in $E^{m+1}$ bounds a compact connected $(m+1)$-manifold?

I am reading Sheila Carter and S.A. Robertson's paper Relations Between a Manifold and its Focal Set. In this paper, they use the following facts: Any closed $m$-manifold $M$ that can be embedded ...
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What is the natural isomorphism between the tangent space of a product manifold and the product of the tangent spaces?

Let $M$ and $N$ be smooth manifolds and $p\in M$, $q \in N$. What is then a natural isomorphism for $$T_{(p,q)}(M\times N) \cong T_pM \times T_pN ?$$
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Classification of contractible 4-manifolds

Is there a general homeomorphism classification of contractible topological 4-manifolds (possibly with boundary or noncompact)? In the compact case, any such manifold has a homology 3-sphere as its ...
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How do I show $X_{\omega(Y,Z)}=-[Y,Z]$?

How do I show $X_{\omega(Y,Z)}=-[Y,Z]$, where $\omega$ is a symplectic 2 form (in particular non-degenerate) and $Y,Z$ are vector fields and $X_f$ is the vector field correspond to the 1 form $df$ ...
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Topology on the affine space of connections

What is the natural topology we generally define on the Affine space of Connections? I am not able to find any literature where this topology is explicitly described. It would be really ...
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Lower bound of embedding dimension for finite CW-Complex of dimension $d$

Consider a finite CW-Complex $C$ of dimension $d$. Let $n$ be the smallest integer such that the complex embeds in $\mathbb{R}^n$. From Whitney it follows that $n \leq 2d$. How would you bound the ...
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On the proof of the Whitney trick (from Scorpan's book)

I'm trying to study a proof of the h-cobordism theorem from Scorpan's "The wild world of 4-manifolds". Given a handle decomposition for the cobordism, the Whitney trick is used to eliminate every pair ...
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Is there a coordinate free( manifestly invariant) expression for the riemann metric tensor on the 2 sphere?

A metric is just a section of the tensor product of the cotangent bundle with itself, a smoothly varying inner product on tangent spaces. . It is a geometric object. I am asking for a geometric ...
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What is the intrinsic ( without any reference to embedding ) and coordinate free , basis free definition of a 2 sphere as a differentiable manifold?

The surface of such an object would be $\dfrac{C^2}\pi$, where $C$ is the circumference, which should be derivable from the definition. If defined as a Riemann manifold, what would it's metric be? (A ...
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Question about critical point of function on compact manifold

How can I deduce that $f$ only has finitely many non-degenerate critical points by this function only has non-degenerate critical point? And how can I use the compact manifold’s properties to solve ...
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A definition for a regular subset of $\partial \Omega$

In the picture below, I'm having some difficulty in understanding the 2nd condition. Why does the second condition make $\Gamma$ lie on one side of its relative boundary? If $\Phi_p$ flattens $\...
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Piecewise linear structure as generators of $C(M)$

Let $M$ be a compact topological manifold and $C(M)$ the commutative unital $C^*$-algebra of complex valued continuous functions on $M$. Suppose that $M$ admits a piecewise linear structure. In the ...
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Is the Hairy Ball Theorem equivalent to saying that the Hopf Fibration has no global sections?

The Hairy Ball Theorem states that $S^2$ has no nonvanishing tangent vector fields. But if we did have such a field then we could normalise each vector so that it lay on the unit circle of the tangent ...
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A relatively open subset of $\partial \Omega$ with a relative boundary.

I'm having some trouble in understanding the following passage of text, specially the part of $\Gamma_1$ and $\Gamma_2$... Also, is $\Omega$ really to be the spherical shell of the sphere, or is ...
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General tips for showing that subspaces “vary continuously”

Here is (part of) a problem from Spivak's Differential Geometry Vol. 1. 6. For a bundle map $(\tilde f,f),$ with $f:B_1\to B_2,$ let $K_p$ be the kernal of the map $\tilde f|_{\pi_1^{-1}(p)}$ from $...
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1answer
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If $n>m$, then the spheres $S^n$ and $S^m$ are not homeomorphic?

We have the following classical result: If $n>m$, then the 1-spheres $S^n$ and $S^m$ are not homeomorphic. Can we prove that with the Jordan-Brouwer theorem? In particular, we set $S^m= (S^m, 0,...
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1answer
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Euler characteristic expression in terms the number of fixed points of an $\mathbb{S}^1$ action

I have found in a paper* that I am reading that Given $(M,J)$ compact (smooth) manifold with an almost complex structure $J$, if we have an $\mathbb{S}^1$ action with isolated fixed points then $ \...
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Singularity of the derivative of a local homeomorphism

If I have a local homeomorphism $D$ defined in the universal covering of the deleted disk $\mathbb{D}^*\subset\mathbb{C}$ has finite degree in a fundamental domain, then 0 is a pole of $f = D'$. I ...
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Elementary reference for existence of a tubular neighborhood of a $C^2$-curve (or elementary proof)?

Consider a 1-periodic function $\varphi \in C^2(\mathbb R;\mathbb R^n)$ with non-vanishing derivative and such that $\varphi|_{[0,1)}$ is a bijection. Let $$ N:=\{\varphi(0)+z\,|\,z\in\mathbb R^n, \, ...
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Intersection number via tangent spaces

Assume that finite groups $G_1$ and $G_2$ act smoothly on a manifold $M$ in such a way that the fixed point set, $M^{G_1\cap G_2}$, is an oriented closed manifold, $M^{G_1}$ and $M^{G_2}$ are its ...
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Local triviality for the fiber bundles

The notations are as follows: \begin{align*} & \operatorname{Diff}^+(\mathbb{D}^2):= \{ f:\mathbb{D}^2\to \mathbb{D}^2\ |\ f \text{ is an orientation preserving diffeomorphism} \},\\ & \...
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An example of smooth but not Riemannian [duplicate]

I've been trying to understand the difference between the notion of smooth (which I understand well) and Riemannian (which I am newly acquainted with). The definition in the tag for 'riemannian-...