# 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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### Smooth Morse function from Forman's discrete Morse function

Let $M$ be a smooth manifold and $K$ a triangulation of $M$, so $K$ is a regular CW-complex and in particular a simplicial complex. Assume that $M$ is compact so $K$ is finite. Let $f\colon K \to \...

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### What is the current state of research in Chern-Simons theory?

I'm a PhD student in mathematical physics looking for some orientation. As asked in the title, I would like to know the current state of research in Chern-Simons theory. More specifically, what are ...

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103 views

### Anti-symmetric operators for the Dirac or Majorana spinors

In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) ...

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239 views

### Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions? [closed]

This is a cross-post of this MSE post that users commented that it is appropriate for MO.
I want to know
Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological ...

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52 views

### Smoothening pseudo-Anosov flows

A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can ...

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69 views

### Reference request: complete, rigorous proof of compactification of moduli spaces of flow lines in Morse homology?

The result I'm looking for can be stated as follows (taken from Hutchings' notes):
Here the moduli spaces are referring to the spaces of flow lines of the negative gradient flow induced by the Morse ...

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91 views

### Commutative square up to sign as indicated, Poincare duality

Let $M$ be smooth oriented manifold with boundary $\partial M $,
${\rm dim}\,M=n$. The two short exact sequences in de Rham cohomology and singular homology
$$0\longrightarrow{}\Omega^{*}(M, \...

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146 views

### Fundamental group of the complement of the arrangement of plane nodal curves

I want to calculate the fundamental group of the complement some collection of plane curves (specifically, two nodal cubics in a general position).
I've read about Severi problem (solved by Harris), ...

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43 views

### Splitting formulas for spectral flows

I'm asking if there are splitting formulas for equivariant spectral flow and higher spectral flow (of Dai-Zhang) for paths of Dirac operators, concerning gluing together two smooth compact Spinc ...

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147 views

### Hopf fibration extended to bundle over $\mathbb{C}^2$

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...

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72 views

### Restriction of diffeomorphisms homotopic to identity to the boundary

Let $M$ be a smooth manifold with boundary $\partial M$. Let $Diff_0(M)$ be the group of all diffeomorphisms homotopic to identity. According to this article (Page 6, section "
Beyond mapping class ...

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50 views

### Spherical space-form as the boundary of an Euclidean ball

Let $M^n$ be a smooth compact manifold such that the boundary $\partial M$ is diffeomorphic to a spherical space-form $S^{n-1}/\Gamma$, where $\Gamma \subset O(n)$ is a finite subgroup acting freely ...

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293 views

### Is there a diffeomorphism of the disk with constant sum of singular values?

This question is a relaxed version of this question.
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $c \ge 2$.
Does there exist a diffeomorphism $f:D \to D$ with constant sum of ...

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55 views

### Local contractibility of group of symplectomorphisms for open manifolds

It is well know that for a closed symplectic manifold $(M, \omega)$ the group of symplectomorphisms in locally contractible. The gist of this proof goes as follows.
Given a $\psi \in \operatorname{...

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54 views

### Is being a deg 0 vector field equivalent to being locally non-surjective?

Let $X:\mathbb R^n\to\mathbb R^n$ be a $C^1$ (or smooth)-vector field, such that $X(0)=0$ is an isolated zero. So we can talk about the mapping of $0$ for $X$.
For convenience assume $0$ be the only ...

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110 views

### Difference between the diffeomorphism classification of a manifold $M$ and the set of equivalences of homotopy smoothings $hS(M)$

In Lopez de Medrano "Involutions on manifolds", a homotopy smoothing of a Poincaré space $X$ is a homotopy equivalence $f:M^n\rightarrow X$, where $M^n$ is a smooth $n$-dim. manifold (everything is ...

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115 views

### Unknotting algorithm in higher dimensions?

Suppose we are given a 2-knot (say by a movie). Is there an algorithm to tell if it is unknotted ? I suppose that it could matter if I say "topologically" or "smoothly" here since those could be ...

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215 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 ...

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75 views

### Diffeomorphisms fixing origin and boundary

Let $D^n$ be a disc in $\mathbb{R}^n$. Is there a known characterization of all the diffeomorphisms of $D^n$ fixing the origin and boundary of $D^n$?

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### Generalizations of Sard-Smale Theorem

Sard-Smale theorem holds for Fredholm maps $f:M\rightarrow B$ between separable Banach manifolds $M,N$. There are some constrains relating the Fredholm index $\operatorname{ind}(f)$ of $f$ to its ...

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51 views

### Smooth involutions on homotopy 11-spheres or diffeomorphism classification of homotopy projective 11-space

Does anyone know if smooth fixed point free involutions on homotopy 11-spheres have been studied? Or equivalently, is something known about the diffeomorphism classification of homotopy $\mathbb{R}P^{...

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59 views

### About the regularity of Thom's first isotopy theorem

Consider an abstract stratified set $(V, \Sigma)$ in the sense of Thom-Mather
(see Mather's note page 491-492 https://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01383-6/S0273-0979-2012-01383-...

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83 views

### Relation between compact vertical cohomology and local cohomology groups

I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt:
The Thom isomorphism in Bott & Tu is obtained as $H_{cv}^{*+n}(E)\rightarrow H^*(M)$, ...

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63 views

### Tangent space to subspace of orbit in jet spaces

I consider a map germ $f: (\mathbb{R}^n,0) \to \mathbb{R} $ which is $k$-determined for some $k \in \mathbb N$, i.e. for all map germs $g: (\mathbb{R}^n,0) \to \mathbb{R} $ having the same $k$-jet as $...

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71 views

### Characterization of extrinsic distance prevserving embedding (see the definition given!) from low dimensional Euclidean spaces to high dimensions

P.S. I asked the question on MSE more than a week ago, but didn't get any desired answer, so asking here.
Let $m < n \in \mathbb{N}$. Let us equip $\mathbb{R}^m, \mathbb{R}^n $ with their ...

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152 views

### About submersion and sections

Let $\pi:X \rightarrow Y$ be a surmersion (surjective submersion) between closed manifolds.
1) Is there any obstruction to the existence of a "multi-valued" section $s$ of $\pi$ such that $\pi \circ ...

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54 views

### Parallel transport of vector along piecewise smooth loop on high-dimensional manifold

In this https://math.stackexchange.com/questions/2568300/gauss-bonnet-like-statement-connecting-parallel-transport-and-curvature question, it was discussed that the rotation of a vector that is ...

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155 views

### Can every diffeomorphism be rescaled into a volume preserving one?

This is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $f:D \to D$ be a diffeomorphism.
Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an ...

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111 views

### Clarify formula for Steifel-Whitney (Poincaré dual) homology classes in a barycentric subdivision?

Let $X$ be a triangulated manifold of dimension $n$. Let $[W_{n-p}] \in H_{n-p}(X,\mathbb{Z}_2)$, be the homology class that's Poincaré dual to the $p$-th Stiefel-Whitney class $[w_p] \in H^p(X,\...

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369 views

### Sard's theorem and Cantor set

Sard's famous theorem asserts that
Theorem. The set of critical values of a smooth function from a manifold to another has Lebesgue measure $0$.
I am asking for the curiosity that is it possible to ...

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153 views

### Elementary questions about vanishing cycles and emerging cycles

Let $X\to D$ be a proper $C^\infty$ map with $D$ an open disk about the origin in some Euclidean space. Suppose $0\in D$ is the only singular value, i.e that over $D^\times=D\setminus \left\{ 0 \right\...

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84 views

### Chart in $1$-parameter family of Lagrangians in a Kähler manifold

Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed real-analytic Lagrangian submanifold. Furthermore, let $L_{t}...

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### Defining the cospecialization in topology

Below is an excerpt from part V of Deligne's Étale cohomology - starting points.
Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the ...

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### The set of embeddings is open in the strong Whitney topology

In Hirsch's book "Differential Topology," he claims in Chapter 2, Theorem 1.4 that the set of $C^1$-embeddings is open in the strong Whitney topology $C^1(M, N)$ where $M$ and $N$ are $C^1$ manifolds. ...

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### Obstruction to the existence of a globally defined integrating factor

Let $U$ be an open subset of $\Bbb{R}^n$ and take $\omega$ to be a nowhere-vanishing smooth $1$-form on $U$. The Frobenius Theorem implies that, near each point of $U$, $\omega$ may be written as $g\,{...

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316 views

### Every _______ $d$-manifold has an $S$-structure

I am looking for some analogous nontrivial but known statements and references about statements of the form:
Every _______ $d$-manifold has an $S$-structure.
Here _______ is a placeholder for ...

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77 views

### $\mathbb{Z}_2$-grading by Hodge star operator (for signature theorem)

This question may be a bit low level for MO but I have not received any attention from the SE post.
Consider the algebra of exterior forms $\bigwedge T^*M$ on an even dimensional $n$-manifold $M$. We ...

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449 views

### Is there an area-preserving diffeomorphism of the disk which is nowhere conformal?

This question is a cross-post; it is related to this former question of mine. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk.
Does there exist a smooth volume-preserving diffeomorphism $f:...

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**1**answer

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### Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves

Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the ...

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237 views

### Density of continuous functions to interior in set of all continuous functions

Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...

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55 views

### Quasi Riemannian submersion and retraction

Let $M, N$ be Riemannian manifolds. A smooth submersion $f:M \to N$ is called a quasi Riemannian submersion if for every $x\in M$ the restriction of linear map $Df_x$ to orthogonal complement of $\...

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161 views

### Riemannian submanifolds of Euclidean space admitting Lipschitz extension of Lipschitz functions, and converse statement

Let $M \subset \mathbb{R}^p$ be a Riemannian submanifold. In what follows, when we talk about a Lipschitz function $f$ on $M$, namely $f: M \to \mathbb{R}$, we will assume there is a $L > 0$ so ...

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217 views

### Shrinking and stretching of vector bundles

Let $M$ be a manifold, $p:E\to M$ a rank $d$ vector bundle. Suppose that $U \subset E$ is an open subset such that $U \cap p^{-1}(x)$ is nonempty and convex for all $x \in M$. Is it true that $U \to M$...

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55 views

### Minimal radius of a ball admitting a trivialization of a vector bundle

Let $X$ be a compact Hausdorff space and $p : V \to X$ a complex vector bundle of rank $n$. For $r > 0$ let $B(r,x)$ denote the open ball of radius $r$ around $x$. Does there exist an $r$ such that,...

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103 views

### Controlling the intersection of two surfaces in $\mathbb{R}^3$

Let $F_1,F_2$ be two closed orientable surfaces embedded in $\mathbb{R}^3$ with genus $2g_1, 2g_2$, respectively (edit: with $g_1, g_2 \geq 1$). Is it possible to isotope around $F_1$ and $F_2$ so ...

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159 views

### Smoothability of open 4-manifolds

F. Quinn proved that any open topological 4-manifold admits a smooth structure in Ends of maps III: dimensions 4 and 5.
He first proves the generalized annulus conjecture:
Suppose $h:D^j\times \...

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243 views

### Is this subset of matrices contractible inside the space of non-conformal matrices?

Set $\mathcal{F}:=\{ A \in \text{SL}_2(\mathbb{R}) \, | \, Ae_1 \in \operatorname{span}(e_1) \, \, \text{ and } \, \, A \, \text{ is not conformal} \,\}$, and
$\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \...

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2k views

### Conceptual proof of classification of surfaces?

Every compact surface is diffeomorphic to $S^2$, $\underbrace{T^2\#\ldots \#T^2}_n$, or $\underbrace{RP^2\#\ldots \#RP^2}_n$ for some $n\ge 1$.
Is there a conceptual proof of this classification ...

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309 views

### On functional equation $f\circ \exp=\exp \circ Df$ on a Riemannian manifold or a Lie Group

Let $M$ be a Riemannian manifold or a Lie group whose corresponding exp map (in corresponding context) is denoted by "exp" which is a map $\exp:TM\to M$
We search for the set $\mathcal{H}...

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472 views

### Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?

This is a cross-post. While working on a variational problem, I have reached to the following question.
Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$...