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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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
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soft question, proof sketch: Constructing spaces of analytic sections. silva spaces

This is a follow up on this question Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations to add more details. In the linked question I ...
53Demonslayer's user avatar
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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
301 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
332 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
226 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
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2 votes
1 answer
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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
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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
131 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
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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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13 votes
1 answer
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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 $2$ ...
Zhenhua Liu's user avatar
3 votes
0 answers
182 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 ...
nick5435's user avatar
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}^...
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12 votes
1 answer
604 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 $F|_{\mathbb{S}^n}=f$...
Piotr Hajlasz's user avatar
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1 answer
151 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 ...
RKS's user avatar
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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 ...
Cuspidal Coffee's user avatar
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 ...
Philip Boyland's user avatar
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0 answers
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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}...
F. Müller's user avatar
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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 ...
GeometriaDifferenziale's user avatar
1 vote
0 answers
124 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 ...
DrHAL's user avatar
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5 votes
1 answer
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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 ...
Radeha Longa's user avatar
3 votes
1 answer
129 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 ...
Learning math's user avatar
0 votes
1 answer
149 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 ...
swami's user avatar
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0 votes
1 answer
335 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 ...
zeta's user avatar
  • 337
11 votes
1 answer
681 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 $g: M\rightarrow N$...
GSM's user avatar
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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 ...
Henry's user avatar
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6 votes
2 answers
341 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, ...
zeta's user avatar
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1 vote
0 answers
125 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 ...
zeta's user avatar
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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 ...
NicAG's user avatar
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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) $ ...
Luis Yanka Annalisc's user avatar
2 votes
0 answers
105 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 ...
Tommaso Rossi's user avatar
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0 answers
67 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 ...
UVIR's user avatar
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6 votes
1 answer
235 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 ...
Aidan Backus's user avatar
4 votes
1 answer
146 views

Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces

I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
ಠ_ಠ's user avatar
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0 votes
2 answers
310 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 ...
Hao S's user avatar
  • 161
1 vote
1 answer
169 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 ...
li ang Duan's user avatar
4 votes
1 answer
420 views

Detecting a "bad map" in Fintushel-Stern knot surgery

Background Let $X$ be a simply-connected smooth 4-manifold which contains a smoothly embedded torus $T$ with trivial normal bundle (in other words, $T^2\times D^2\subset X$). Let $K$ be a knot in $S^3$...
rab's user avatar
  • 139
3 votes
0 answers
208 views

Topologies on diffeomorphisms groups

Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual ...
HUO's user avatar
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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, ...
Chicken feed's user avatar
4 votes
1 answer
167 views

Version of pseudo-isotopy $\neq$ isotopy for $(n+1)$-framings

Let $M$ be a closed $n$-manifold and $\varphi$ be a self-diffeomorphisms of $M$. There is a bordism from $M$ to itself given by $M\times [0,1]$ with the identification $M \cong M \times \{0\}$ induced ...
Daniel Bruegmann's user avatar
16 votes
0 answers
390 views

Is the oriented bordism ring generated by homogeneous spaces?

I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
Zhenhua Liu's user avatar
6 votes
0 answers
199 views

"Inclusion" between higher categories of framed bordisms?

Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds. It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences. If $n$ is large enough, ...
Daniel Bruegmann's user avatar
5 votes
1 answer
328 views

To what extent differentiable mappings of an affine line into a manifold determine its differentiable structure? What about mappings of a plane?

If $M$ is a (real) differentiable manifold, its differentiable structure is completely determined if it is known which mappings $M\to\mathbf{R}$ are differentiable. How much can be said about the ...
Alexey Muranov's user avatar
18 votes
1 answer
2k views

Intuition behind manifolds which are homeomorphic but not diffeomorphic

Popular articles on mathematics often explain the difference between homeomorphism and diffeomorphism with statements like - "A rectangle is homeomorphic to the circle but not diffeomorphic to it&...
Anindya's user avatar
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21 votes
3 answers
2k views

Is every linear functional on a smooth finite dimensional vector space automatically smooth?

By a smooth finite dimensional vector space I mean a smooth manifold $M$ together with smooth operations $+ : M \times M \rightarrow M$ and $\cdot : \mathbb{R} \times M \rightarrow M$ turning $M$ into ...
Tristan Bice's user avatar
  • 1,327
1 vote
0 answers
40 views

Verdier (w) condition implies the $w_f$ condition when the restriction of $f$ in each stratum is a submersion?

Let $X\subset\mathbb{R}^n$ be and let $\Theta=(X_\beta)_{\beta\in I}$ a Verdier stratification for X. Let $f:X\rightarrow\mathbb{R}$ be a polynomial function, such that $f_{|_{X_\beta}}$ is submersion ...
Giovanny Snaider Barrera Ramos's user avatar
0 votes
0 answers
113 views

Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$. On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...
gpr1's user avatar
  • 134
9 votes
0 answers
128 views

degree 1 maps for bordism homology

Let $f\colon X \to Y$ be a degree 1 map between closed oriented manifolds. Then the induced homomorphism between the homology groups is surjective up to torsion. Can one say something similar about (...
Klaus Niederkrüger's user avatar
1 vote
0 answers
130 views

Higher dimensional Seifert surfaces and link numbers of higher knots

In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots. Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
0x11111's user avatar
  • 473
5 votes
0 answers
121 views

Division of fibration by $\Sigma_{n}$ gives Serre fibration

This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
TopologyStudent's user avatar

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