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”.
1,741
questions
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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
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1
answer
119
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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
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42
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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 $...
1
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0
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68
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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 ...
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1
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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
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1
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338
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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
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1
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233
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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
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1
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184
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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 ...
2
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0
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158
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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 ...
5
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134
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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 ...
13
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321
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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) ...
13
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1
answer
472
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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 $2$ ...
3
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183
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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 ...
3
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1
answer
314
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"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}^...
12
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1
answer
614
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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 $F|_{\mathbb{S}^n}=f$...
0
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1
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154
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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
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61
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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 ...
1
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1
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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
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69
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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}...
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1
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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 ...
1
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0
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125
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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 ...
5
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1
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387
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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 ...
3
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1
answer
130
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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 ...
0
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1
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150
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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 ...
0
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1
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340
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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
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1
answer
686
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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 $g: M\rightarrow N$...
4
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239
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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
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"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
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0
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126
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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
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0
answers
108
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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
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2
answers
412
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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
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0
answers
107
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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 ...
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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 ...
6
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1
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237
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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 ...
4
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1
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150
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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 ...
0
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2
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316
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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 ...
1
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1
answer
171
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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 ...
4
votes
1
answer
422
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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 $S^3$...
3
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213
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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 ...
5
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195
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$C^1$ 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, ...
4
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1
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167
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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 ...
16
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0
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391
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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 ...
6
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0
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199
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"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, ...
5
votes
1
answer
329
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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 ...
18
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1
answer
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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&...
21
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3
answers
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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 ...
1
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0
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40
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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 ...
0
votes
0
answers
116
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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, ...
9
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0
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130
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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 (...
1
vote
0
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132
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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 $...