# 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”.

1,781
questions

2
votes

1
answer

188
views

### 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 ...

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}\...

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 ...

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 ...

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 \...

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 ...

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 ...

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 ...

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 $\psi_1, \psi_2, \cdots, \psi_n$, ...

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{...

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{...

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 ...

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 ...

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 ...

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 ...

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 ...

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(\...

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 \...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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?

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 ...

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 ...

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=[-\...

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 ...

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 ...

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 (...

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 $\...

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 ...

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 ...

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)...

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?

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-...

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 ...

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 $...

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 ...

-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 ...

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, ...

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 ...

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 ...

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 ...

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 ...

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) ...