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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-1 votes
0 answers
100 views

Prove a generalization of hairy-ball theorem

I found an interesting question below: Prove that the 6-sphere admits no continuous field of tangent 3-planes.
0 votes
1 answer
118 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 ...
9 votes
1 answer
501 views

Submersion vs fiber bundle

If one starts with a fiber bundle $f: X \to Y$ so that fibers having trivial integral homology by using spectral sequence one can get the induced map $f_*: H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is ...
2 votes
0 answers
42 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
1 answer
102 views

Existence of a Hölder homeomorphism satisfying prescribed norm constraints

Let $\Omega$ be a convex body$^{\boldsymbol{1}}$ in $\mathbb{R}^n$ where $n$ is a positive integer. Fix a positive integer $k$ and some $0<\alpha\leq 1$. Let $k_1> k_2>0$. Does there ...
9 votes
1 answer
349 views

Projective span of a manifold

Recall that the span of a smooth manifold $M$, denoted $\operatorname{span}(M)$, is the largest $k$ such that $M$ admits $k$ linearly independent vector fields. Equivalently, $\operatorname{span}(M)$ ...
1 vote
0 answers
68 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 ...
8 votes
2 answers
790 views

A relative version of Ehresmann's theorem

Edited: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again. Let $N\subset M$ be a pair ...
-4 votes
1 answer
303 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
337 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
184 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 ...
9 votes
1 answer
615 views

Does the continuous mapping space between topological manifolds always admit a Banach manifold structure?

Let $M$ and $N$ be smooth, i.e. $C^\infty$, manifolds. Suppose that $M$ is compact. Then for every $k \geq 0$ it is well known that $$C^k(M,N)$$ admits the structure of a smooth Banach manifold. I am ...
2 votes
1 answer
233 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 ...
2 votes
1 answer
109 views

Minimal dimension for immersions to be dense in the continuous function space

Let $f:[0,1]^n \rightarrow \mathbb{R}^m$ be an arbitrary continuous function. My question is, under what conditions for $m$, there exists an immersion $g:(-\epsilon,1+\epsilon)^n \rightarrow \mathbb{R}...
2 votes
0 answers
158 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
134 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 ...
5 votes
3 answers
664 views

Counting connected manifolds

Apparently, there is the following fact: The set of homeomorphism classes of connected manifolds has the same cardinality $c$ as that of $\mathbb R$. I find it to be interesting; but would be ...
13 votes
0 answers
321 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) ...
13 votes
1 answer
472 views

Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces

Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
3 votes
1 answer
314 views

"Totally real" linear transformations

Identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ via the equality $$(z_1, z_2, \ldots, z_n)=(x_1, \ldots, x_n, y_1, \ldots , y_n)$$ Where $z_j=x_j + iy_j$. We call a linear invertible map $A: \mathbb{R}^...
3 votes
0 answers
183 views

Reference for a folklore theorem about h-cobordisms

I've seen referenced here that if $M$ and $N$ are closed topological $n$-manifolds and $f: \mathbb{R}\times M \to \mathbb{R}\times N$ is a homeomorphism, then $M$ and $N$ are h-cobordant. I know that ...
2 votes
1 answer
166 views

Leaf holonomy of Reeb foliation on mobius strip

I am trying to understand the leaf holonomy of the Reeb foliation on the mobius strip, the first problem being visualization. I have been unable to find a visualization of this anywhere. I am ...
12 votes
1 answer
614 views

Isotopic diffeomorphisms of the sphere

Assume that $f:\mathbb{S}^n\to\mathbb{S}^n$ is a diffeomorphism and assume that there is an orientation preserving diffeomorphism $F:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ such that $F|_{\mathbb{S}^n}=f$...
1 vote
1 answer
72 views

When is the real Abel-Jacobi-Albanese map injective?

$ \def\tMA{\tilde{M}_a} \def\tomega{\tilde{\omega}} \def\tx{{\tilde{x}}} \def\tzeta{\tilde{\zeta}} \def\T{{\mathbb T}} \def\R{{\mathbb R}} \def\Z{{\mathbb Z}} \def\raw{\rightarrow} $ I want to work ...
0 votes
1 answer
154 views

Implicit function theorem and its consequence

Let $f:{\Bbb C}^{n+1}\to {\Bbb C}^n$ be a map defined by homogeneous polynomials. There is a point $p\in f^{-1}(0)$ and a neighbourhood $U$ of $p$ in $f^{-1}(0)$, such that $d(f)$ has rank $n$ at ...
2 votes
0 answers
61 views

Continuous invariants of singularities in the Thom-Mather theory of deformations

I have been reading through Arnold et al.'s Singularities of differentiable maps to have an understanding on Arnold's theory of deformations of wave fronts. His theory is similar to the Thom-Mather ...
3 votes
1 answer
147 views

Convex hull and least area discs in Riemannian 3-manifolds

$\DeclareMathOperator\Conv{Conv}$Let $M$ be a complete Riemannian 3-manifold and $\gamma \subset M$ a simple closed curve that bounds a least-area disc $D$ - a disc that minimizes the area among all ...
1 vote
1 answer
192 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
0 votes
0 answers
69 views

Existence of covering space with trivial pullback map on $H^1$

I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
0 votes
1 answer
108 views

Do covector fields correspond to homomorphisms of $ \mathscr C^\infty $-modules from the sheaf of vector fields to the sheaf of smooth functions?

This question has been crossposted from MSE since there it received no attention. Please notify me if questions like these are not appropriate for this platform. The question Let $ M $ be a smooth ...
5 votes
1 answer
387 views

A question about the existence of spin maps

Let $M, N$ be two smooth manifolds, not necessarily spin. My question is the following: How can we construct a non-constant spin map $f:M\to N$ of degree zero? Here spin map means that $f$ preserves ...
1 vote
0 answers
125 views

Poincaré-Hopf Theorem for domains with a point of vanishing curvature

Consider $\Omega \subset \mathbb{R}^2$ a convex planar domain having positive curvature on the boundary except for a point $p \in \partial \Omega$ where the curvature vanishes. I would like to know ...
3 votes
1 answer
130 views

Homogeneous regular (= polynomial component) maps with odd degree and their being global homeomorphisms in dimensions higher than one?

Let $F:\mathbb{R}^m \to\mathbb{R}^m, F:=(F_1\dots F_m)$ be a regular map, i.e. with components $F_i$ that are polynomials. Assume further that each $F_i$ is an odd degree (say $d$) homogenous ...
12 votes
4 answers
2k views

Fundamental groups of compact Kähler manifolds

This is a sort of a follow-up to this question, and especially to Sean Lawton's answer: The book Fundamental Groups of compact Kähler manifolds (which, in my opinion, is one of the best mathematics ...
0 votes
1 answer
340 views

Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]

We know that framing structure means the trivialization of tangent bundle of manifold $M$. string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
11 votes
2 answers
1k views

Thom's first isotopy lemma

Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that $f|_S:S\to P$ is proper and $f|_X:X\to P$ is a ...
0 votes
1 answer
150 views

Formulating a 3D "twist" transformation for a unit circle into a lemniscate while preserving arc length

I'm exploring the transformation of a 2D unit circle into a lemniscate (infinity symbol) by fixing two antipodal points and "twisting" the circle (in the 3rd dimension) such that the ...
11 votes
1 answer
686 views

Smooth map between oriented manifolds

Let $f: M\rightarrow N$ be a smooth map between smooth closed oriented connected manifolds of same dimension. Question: is it true that $f$ is smoothly homotopic to some smooth map $g: M\rightarrow N$...
15 votes
2 answers
3k views

Measures and differential forms on manifolds

Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$. What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of ...
4 votes
0 answers
239 views

Normalizer of the group of segment $C^\infty$ diffeomorphisms in the group of segment homeomorphisms

What is the normalizer of the group of $C^\infty$ diffeomorphisms on $[0, 1]$, with group law given by composition, in the group of all homeomorphisms of $[0, 1]$? If the answer is known, is there ...
6 votes
2 answers
343 views

"canonical" framing of 3-manifolds

In Witten's 1989 QFT and Jones polynomial paper, he said Although the tangent bundle of a three manifold can be trivialized, there is no canonical way to do this. So if I understand correctly, ...
1 vote
0 answers
126 views

Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character

In Witten's 1989 QFT and Jones polynomial paper, he wrote in eq.2.22 that Atiyah Patodi Singer theorem says that the combination: $$ \frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi} $$ is a ...
2 votes
0 answers
108 views

Progess on conjectures of Palis

I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures "Global Conjecture: There is a dense set $D$ of dynamics such that any element of ...
3 votes
2 answers
412 views

A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $

Consider a manifold $ N $ defined as follows $$ N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3}, $$ where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
2 votes
0 answers
107 views

Extension of isotopies

In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...
0 votes
2 answers
316 views

If a graph embedded on a surface is divided by a curve into a right and left that do not intersect can it be embedded on a surface of smaller genus?

Suppose we have a graph $G$ embedded on a (smooth, orientable etc) surface $Q$. Suppose there is a cycle $C$ of $G$ such that $C$ does not separate our surface $Q$ into two connected regions and ...
0 votes
0 answers
68 views

Topological transversality by dimension

We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
5 votes
0 answers
195 views

$C^1$ manifold with complex structure

Let $M$ be a manifold. A complex structure on $M$ is an endomorphism $J \in \text{End}(TM)$ such that $J^2 = -\text{id}$ together with the vanishing of the Nijenhuis tensor. If $J$ is real-analytic, ...
1 vote
1 answer
171 views

Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?

Because flowmaps are homeomorphic maps, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system? that is, does ...
6 votes
1 answer
237 views

Can differential forms be exact and positive on a distribution?

Let $M$ be a manifold of dimension $d$, and let $\mathscr D$ be a distribution of rank $d - 1$ on $M$ (I would also be interested in lower rank distributions, but mainly I am interested in codimension ...

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