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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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No analytic surjection $f:M \to N$ when $\dim(M) >\dim(N)$

Inspired by comment discussions in this MO post smooth version of splitting principle we ask: Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any ...
Ali Taghavi's user avatar
1 vote
0 answers
70 views

A counterexample to the extendibility property for submanifolds

Let $(M,g)$ be a Riemannian manifold. We say that a bounded embedded submanifold $S\subset M$ of class $C^k$ has the extendibility property if there exists a larger embedded submanifold $\tilde{S}\...
trenghia's user avatar
0 votes
0 answers
57 views

Is there apecial conditions to be if Tangent space of y over itself is contained in M over y, then y is in M?

Here are two theorems from Hirsch's book: 4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ map. If $y \in N - \partial N$ a ...
Snailman's user avatar
1 vote
0 answers
195 views

Smooth version of the splitting principle

Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
Ali Taghavi's user avatar
12 votes
0 answers
219 views

When can we extend a diffeomorphism from a surface to its neighborhood as identity?

Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
Anubhav Mukherjee's user avatar
2 votes
0 answers
145 views

A question regarding weak Whitney embedding theorem

The weak Whitney embedding theorem states that any continuous map $f: D^n \to \mathbb{R}^{2n+1}$ (Let us focus on $D^n$ for this question) can be approximated (in $C^0$-norm) by embeddings. A counter ...
Rancho's user avatar
  • 21
12 votes
0 answers
289 views

Is there a differential form which corresponds to an eigenvalue of the homomorphism in cohomology?

Let $M$ be a closed manifold and $f:M\to M$ be a diffeomorphism. Suppose the homomorphism $f^*:H^k(M;\mathbb R)\to H^k(M;\mathbb R)$ has an eigenvalue $\lambda\in\mathbb{R}$. Note that $\lambda$ is ...
Andrey Ryabichev's user avatar
16 votes
0 answers
554 views

Must the number of smooth structures be countable or continuum?

Let $M$ be a manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails? Edit: By ...
183orbco3's user avatar
  • 431
2 votes
1 answer
127 views

Extending diffeomorphisms between surfaces

Suppose we have two smooth compact oriented surfaces $M_1$ and $M_2$ with boundary,both of them have genus $g$, and there are orientation preserving diffeomorphisms $\psi_1, \psi_2, \cdots, \psi_n$, ...
LDLSS's user avatar
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3 votes
1 answer
242 views

If the complement of a knot $K$ fibers over the circle is $K$ necessarily fibered?

Let $K \subseteq S^3$ be a knot in the $3$-sphere and assume there exists a smooth map $p \colon S^3\setminus K \to S^1$ which is a fiber bundle. For every point $\require{enclose} \enclose{...
Patrick Perras's user avatar
2 votes
0 answers
79 views

Topological constraints on real algebraic surfaces from mean curvature constraints

The mean curvature of a real algebraic surface $S$ in $\mathbb{R}^3$ defined as the zeroes of a polynomial $P$ of degree $d$ with real coefficients in three variables is given by the formula \begin{...
Yromed's user avatar
  • 173
3 votes
0 answers
100 views

Do any Legendrian knots in standard contact 3-space have big tubular neighborhoods?

Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$. According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular ...
Matthew Kvalheim's user avatar
0 votes
0 answers
124 views

What do associated fibre bundles have in common?

Two fibre bundles are said associated if they have isomorphic associated principal bundles. I understand that this means they are defined by the same transition functions, but still is there some more ...
Lefevres's user avatar
0 votes
0 answers
149 views

What is the adjoint bundle of groups $P\times_{G}G$?

It is said that G acts on itself by conjugation. I am familiar with another type of adjoint bundle in which a representation of G on a vector space is given. Can someone explain the differences and ...
Lefevres's user avatar
5 votes
1 answer
267 views

Schoenflies problem in PL setting

What is the status of the Schoenflies problem in the PL category? In other words, given an injective PL map $f:S^{n-1} \hookrightarrow S^n$, is it always PL equivalent to the equatorial inclusion? (I ...
Victor's user avatar
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3 votes
0 answers
74 views

Are these contact structures on the open solid torus diffeomorphic?

Let $M=S^1\times \mathbb{R}^2$ and $\alpha_1, \alpha_2$ be a pair of contact one-forms on $M$ such that the restrictions $\alpha_1|_{S^1\times \{0\}}$, $\alpha_2|_{S^1\times \{0\}}$ coincide and ...
Matthew Kvalheim's user avatar
4 votes
1 answer
330 views

Bott & Tu differential forms Example 10.1

In Bott & Tu's "Differential forms", Example 10.1 states: $\textbf{Example 10.1}$ Let $\pi: E \to M$ be a fiber bundle with fiber $F$. Define a presheaf on $M$ by $\mathcal F(U) = H^q(\...
Jaehwan Kim's user avatar
3 votes
0 answers
99 views

Relation between number of critical points of harmonic functions and number of connected components of the level sets

I am asking what I think is a simple question in the general area of Morse theory, specialized to 2-d and harmonic functions. I'll be specific. Suppose I have $U(z)$ positive and harmonic for $z\in \...
Marc Berth's user avatar
2 votes
0 answers
138 views

Compute the Euler class of tautological $C$-bundle over $CP^1$

$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here. The background is that we need to use a similar ...
threeautumn's user avatar
1 vote
0 answers
55 views

extendability of fibre bundles on manifolds with same dimensions

Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where $M'$ is also an $m$-manifold. Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where $N'$ is also an $n$-manifold. Suppose there is fibre ...
Shiquan Ren's user avatar
6 votes
0 answers
128 views

Are there isospectrally equivalent exotic spheres?

Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum? I would be happy ...
discretephenom's user avatar
8 votes
1 answer
436 views

On the definition of stably almost complex manifold

According to Adams' paper "Summary on complex cobordism", a manifold is stably almost-complex if it can be embedded in a sphere of sufficiently high dimension with a normal bundle which is a ...
onefishtwofish's user avatar
0 votes
0 answers
23 views

Heuristics for constrained maximal volumes in hypercubes as $n \to \infty$

It can be shown that there is a unique maximal surface of revolution with constant positive Gaussian curvature embedded in $[0,1]^3$ with a pair of antipodal points as cone points which attain the ...
53Demonslayer's user avatar
0 votes
1 answer
75 views

On nontrapping manifolds

Suppose that $(M,g)$ is a compact connected smooth Riemannian manifold without boundary. Let $U \subset M$ be a smooth submanifold of codimension zero with smooth boundary and assume that $U$ is ...
Ali's user avatar
  • 4,113
2 votes
0 answers
94 views

lifting a family of curves to a family of sections of a vector bundle?

This is a question in obstruction theory. It should be basic but I can't find a reference. Let's stick to the $C^\infty$ category, so all objects mentioned are smooth. Let $\pi: E \to M$ be a vector ...
skwok's user avatar
  • 21
4 votes
1 answer
285 views

Derivatives of diffeomorphism whose iterates on an open set converge to a point

Consider a smooth manifold $M$, a diffeomorphism $\varphi\in\mathrm{Diff}^\infty(M)$, and an open subset $B\subseteq M$. Suppose that, when restricted to $B$, $\varphi^n$ converges uniformly to a ...
user815293's user avatar
3 votes
1 answer
358 views

Topology and local isometry, spinning cosmic string

Suppose one is given the spacetime $(M,g)$ where $M$ is a fixed differentiable manifold and $g$ is a Lorentzian metric whose local expression is: $$g= -(dt + a \, d \phi)^2 + d\rho^2 + \kappa^2 \rho^2 ...
Bastam Tajik's user avatar
7 votes
2 answers
554 views

Locally conformally flat

Is there any example of a locally conformally flat manifold that is neither a space form nor a product of space forms?
Sayoojya's user avatar
1 vote
0 answers
154 views

If $X$ is a strong deformation retract of $\mathbb{R}^n$, then is $X$ simply connected at infinity?

Let $X \subseteq \mathbb{R}^n$, and assume there is a strong deformation retract from $\mathbb{R}^n$ to $X$. Is $X$ necessarily simply connected at infinity? (Edit) Follow up question: if there is a ...
ccriscitiello's user avatar
2 votes
1 answer
101 views

Finite group extensions of lattices

I'm currently reading the proof of Geroch's conjecture in Lawson-Michelsohn's Spin Geometry book and in the proof of Proposition IV.5.8 that every Ricci-flat enlargeable manifold is flat the following ...
pizzalberto's user avatar
0 votes
1 answer
122 views

Local embedding and disk in domain perturbation

Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
monoidaltransform's user avatar
4 votes
1 answer
260 views

Is the wildness of 4-manifolds related to the diversity of their fundamental groups?

$n = 4$ is the smallest dimension such that the fundamental group of a closed $n$-manifold can be any finitely-presentable group (leading e.g. to various undecidability results stemming from the ...
Tim Campion's user avatar
  • 62.8k
2 votes
1 answer
113 views

Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonnegative curvature operator

Let $S^m$ be a standard sphere of dimension $m=n+4k$, and let $M$ be any closed Riemaninan manifold of dimension $n$ with nonnegative curvature operator. My question: Is there always a smooth spin map ...
Radeha Longa's user avatar
1 vote
0 answers
58 views

Extending the natural thom form of a vector bundle from the boundary of a manifold

(Edited after taking into account Tom Goodwillie's answer.) Let $E \rightarrow X$ be an orientable vector bundle. In this MO answer it is explained how to obtain a representative of the Thom class (...
Kai Hugtenburg's user avatar
5 votes
1 answer
375 views

Linking number and intersection number

Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
user429294's user avatar
13 votes
0 answers
292 views

Is there an analogue of Steenrod's problem for $p>2$?

An element $\alpha \in H_k(X; \mathbb{Z})$ is said to be realisable if there is a $k$-dimensional connected, closed, orientable $k$-dimensional submanifold $Y$ such that $\alpha = i_*[Y]$. The ...
Crash Bandicoot's user avatar
0 votes
1 answer
79 views

Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
MathLearner's user avatar
4 votes
1 answer
166 views

Isotopies of codimension-1 disks relative to boundary

I at first thought this should be an easy question, but then realized it might actually not even be known. Let us work in the smooth category, though I am also wondering about PL and TOP (locally flat)...
Blake Winter's user avatar
2 votes
0 answers
88 views

Differential operators and iterations of tangent bundle

Is there a relationship between higher order differential operators and higher tangent bundle viewed as bundle on the base manifold?
Lefevres's user avatar
2 votes
0 answers
109 views

Are oriented-$h$-cobordant lens spaces orientation-preservingly homeomorphic?

Consider two three-dimensional lens spaces $N_1=L(p,q_1)$ and $N_2=L(p,q_2)$, and assume that there is an oriented-$h$-cobordism between them. In other words, we assume that there is an oriented four-...
Nathan's user avatar
  • 21
0 votes
1 answer
137 views

Vector bundles over a homotopy-equivalent fibration

I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here. Let $\pi:N\rightarrow M$ be a smooth ...
Bence Racskó's user avatar
2 votes
0 answers
45 views

Under what conditions principal directions define an integrable distribution?

Consider a hypersurface $M^n \subset \mathbb{R}^{n+1}$ which is compact without boundary. Assume that its second fundamental form $A$ has distinct eigenvalues $\lambda_1<\ldots<\lambda_k$ (with $...
Dorian's user avatar
  • 331
1 vote
0 answers
70 views

Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
53Demonslayer's user avatar
-4 votes
1 answer
324 views

Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
Bastam Tajik's user avatar
4 votes
1 answer
383 views

Criteria for extending vector field on sphere to ball

Below is a theorem that is equivalent to Brouwer fixed-point theorem, which I found quite interesting. The proof is in this PDF file. Let $v: \mathbb S^{n-1} \to \mathbb R^n$ be a continuous map, ...
Zhang Yuhan's user avatar
2 votes
1 answer
277 views

How to chart tubes around manifolds with boundary/corners?

Let $M \subset \mathbb{R}^d$ be a manifold with boundary/corners. For example, a piece of curve with endpoints or a $2d$ unit square in $\{ z = 0 \}$. I am interested in introducing local coordinates ...
tsnao's user avatar
  • 600
3 votes
1 answer
192 views

Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
Yasha's user avatar
  • 491
2 votes
0 answers
182 views

Classification of bundles with fixed total space

I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
Matthew Kvalheim's user avatar
5 votes
0 answers
148 views

Representing some odd multiples of integral homology classes by embedded submanifolds

Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
Zhenhua Liu's user avatar
14 votes
0 answers
339 views

Nonsmoothable 4-manifolds

Does there exist a closed connected nonsmoothable 4-manifold $M$ such that: $\kappa(M)=0$ (Kirby-Siebenmann invariant vanishes, hence, there is no "classical" obstruction to smoothability) ...
Moishe Kohan's user avatar
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