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”.
1,753
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Classifying smooth embeddings which yield Morse functions
Let $\mu:M \to \mathbb{R}$ be a fixed surjective smooth function on a smooth manifold $M$. Let $N$ be a smooth compact manifold that embeds smoothly into $M$ via $\iota:N \to M$.
What conditions on ...
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1
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Heisenberg group: research themes
I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties.
I would like to know ...
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Is there a general theory of Sard measures?
Let $f:M \to N$ be a smooth map of smooth (second countable) manifolds. The set $C_f \subset M$ of critical points of $f$ is defined to be the set of all $m \in M$ such that the differential $df : T_{...
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Special Morse function on a Riemann surface
Let $f: S \to \Bbb R$ be a Morse function on a Riemann surface. Let $x_0$ be a saddle point of $f$. Since $x_0$ is a critical point of $f$, it makes sense to talk about the bilinear forms $f_{z\...
- 1,171
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3
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Proving the existence of good covers
Usually one proves the existence of good covers in compact manifolds by Riemannian methods: we pick an arbitrary Riemannian metric, prove that geodesically convex neighborhoods exist, that they are ...
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2
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Intersection forms of 4-manifolds with boundary
Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...
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Perturbation of Morse function at a critical point
I recently learned from a knowledgeable person that for a Morse function $f: M \to R$ with a critical point $x_0$, one can perturb $f$ in such a fashion that the new function has the same critical ...
- 1,171
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1
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Diffeomorphisms of a surface in terms of generators.
I am interesting in a presentation of a diffeomorphisms in terms of generators. Is it possible to obtain such presentation in some cases, depending on a genus of a surface or a type of diffeomorphism (...
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3
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Operations via Morse Theory
I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)...
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Kervaire invariant: Why dimension 126 especially difficult?
Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\...
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Natural Transformations from the Tensor product of tangent bundle into the second order tangent bundle
Short question with long title:
Suppose $T$ is the tangent functor and $T^2:=T\circ T$ is the second order tangent functor.
Are there natural transformations $T\otimes T \Rightarrow T^2$ ?
I ...
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Extending psc metrics
Let $S^1$ denote the circle with the non-trivial spin structure, i.e. $0\neq[S^1]\in\Omega^{Spin}_1$. Considering characteristic numbers it is easy so see that $S^1\times\mathbb{H}P^3$ is spin null ...
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1
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Homology and homotopy of a surface
Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H_1(S, \mathbb Z).$
My question is; does this ...
user22741
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finiteness of the dimensions of cohomologies of open subsets of a compact manifold
Let $M$ be a compact differentiable manifold which can be covered by two open subsets $U$ and $V$. Then $H_{\text{dR}}^n(M)$ is finite-dimensional for all $n$. But how about $U$, $V$ and $U\cap V$? ...
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Metric deformations from non-negative to positive curvature
Is it possible to deform the metric $g$ of a closed Riemannian manifold $(M,g)$ satisfying $\mathrm{Ricci}(M,g) > 0$ and $\mathrm{sec}(M,g) \geq 0$ to a metric $g_1$ satisfying $\mathrm{sec}(M,g_1) ...
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Topological version of two results in smooth Morse theory
Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references).
It is well known ...
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How to tackle the smooth Poincaré conjecture
The last remaining problem in this whole "everything is a sphere" business, is the smooth Poincaré conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^...
- 17.2k
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3
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Computing the Euler characteristic of the complex projective plane using differential topology
I am trying to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for ...
- 73
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1
answer
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Are isospectral manifolds necessarily homeomorphic?
It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric.
Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic?...
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representatives of the group of homotopy 7-spheres
In Milnor's paper "On manifolds homeomorphic to the 7-sphere" it is proven that there are manifolds homeomorphic but not diffeomorphic to the standard 7-sphere. His construction involves sphere ...
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1
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The equivariant index of Dirac operator
Let us consider the Dirac complex
\begin{equation}
D_{\rm Dirac}:S^+\to S^-
\end{equation}
where $S^{\pm}$ are the chiral-spinor bundles on $\mathbb{R}^4$.
Using the fact that the bundle $S^+$ is ...
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answer
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Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ for n >1
So first for n even, $RP^n$ is not orientable, hence can not be embedded in $\mathbb{R}^{n+1}$.
For odd n, $RP^{n}$ is orientable, hence the normal bundle is trivial. Now using stiefel-Whitney ...
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1
answer
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Actions of finite groups on exotic smooth manifolds of dimension >4
Let $M_1^n$ and $M_2^n$, $n>4$ be two smooth compact manifolds that are homeomorphic but not diffeomorphic. Suppose that a finite group is $G$ acting faithfully on $M_1^n$ by diffeomorphisms. Is it ...
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Deforming Fredholm sections
Suppose we have a Fredholm section S, (the differential is Fredholm at the 0-set) of some
Banach vector bundle over X, transverse to the 0-section, with Fredholm index 1 and such that the 0-set of ...
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votes
1
answer
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On The Definition of Smoothness in "Differential Topology" by Guillemin & Pollack
This is a question about the definition of a smooth function in Guillemin and Pollack's "Differential Topology". G&P define all manifolds as objects embedded in $\mathbb{R}^N$ for some $N$, which (...
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Embedding of $T^{2}$ on $S^{1}\times S^{2}$.
Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be a embedding map and $i_{*}:\pi(T^{2})\rightarrow \pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times S^{...
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Pullbacks and Inclusions of Smooth function algebras of manifolds.
Let $M$ and $N$ be two smooth finite dimensional manifolds and
$C^\infty(M)$ as well as $C^\infty(N)$ their smooth function algebras.
Is the following true:
Let $\imath: M \to N$ be an embedding. ...
- 2,014
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votes
1
answer
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Handlebody decomposition of an open 4-manifold
Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth ...
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2
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Manifolds with homeomorphic interiors
Suppose that two compact topological manifolds with boundary have homeomorphic interiors. Can we conclude that the two manifolds are homeomorphic? What happens in the smooth category?
- 2,283
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2
answers
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Relationship between double tangent bundle, exterior derivative and connection
I am totally new to the subject differential geometry, and that probably reflects itself in the naive question that I'm trying to formulate. I hope this question does not get closed because of this.
...
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How to compute the Monopole Floer Homology for Surface $\times S^1$ ?
We know that Monopole Floer homology of a 3-manifold $M$ depends on a spin-c structure. My question is that if $M$ is $F\times S^1$ ($F$ is a surface of genus larger than 1) then how can we compute ...
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Relation on the set of connected components of the $\mathbb{C^*}$-fixed points locus coming from the Bialynicki-Birula decomposition
Let $X$ be a smooth variety with an action of $\mathbb{C}^*.$ One has the so-called Bialynicki-Birula decomposition of $X$ given by stable manifolds: $$X=\bigcup_N X^+(N),$$ where $N$ varies in the ...
user17778
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Weil Kostant Integrality Result as Stated by Brylisnki
I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued 2-...
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Does every vector bundle allow a finite trivialization cover?
Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is ...
- 1,264
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Singular fibers of generic smooth maps of negative codimension
This is in some sense a follow-up to my question on submersions.
Let $f\colon\thinspace M\to N$ be a generic smooth map between closed manifolds of dimensions $m$ and $n$. Assume that the codimension $...
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Example of a Sheaf (on the site of smooth manifolds) with Nontrivial Cohomology on $\mathbb{R}^n$?
Does such a sheaf of abelian groups exist? If not, is there a reference or a proof? Does such a sheaf of non-abelian groups exist?
I realized recently that while I've taken it for granted that ...
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3
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How do we use an Ehresmann connection to define a semispray?
Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.
Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a ...
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votes
2
answers
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Submersions of closed manifolds
Let $f\colon\thinspace M\to N$ be a map of closed smooth manifolds, with $\dim M > \dim N$. Recall that a submersion is a smooth map whose differential is surjective at every point in the domain.
...
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Smooth representatives for elements of $\pi_7(\text{exotic $S^7$})$
Let $M$ be $S^7$ with an exotic smooth structure. Since one can smoothen maps, there exist smooth maps $f:S^7\to M$ which are homotopic to the identity (relative to a base point, if you want).
Can ...
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Euler class in the non-compact case
Does anyone have a reference for:
The Euler-class for an open non-compact
manifold possibly with twisted coefficients (if the
group action on the manifold does not preserve
orientation) and/or for a ...
- 3,850
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votes
2
answers
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Do infinite products commute with functor of smooth sections?
Similarly to my previous question about direct limits, I have now basically the same question about inverse limits. It seems in fact, that I only need the result for products.
Question: Is there a ...
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Good introduction to Morse-Novikov theory?
Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued ...
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Recognizing the 4-sphere and the Adjan--Rabin theorem
The problem of recognizing the standard $S^n$ is the following:
Given some simplicial complex $M$ with rational vertices representing a closed manifold,
can one decide (in finite time) if $M$ is ...
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3
answers
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Criteria for Involutive Subbundles
Preliminaries: Let $M$ be a smooth manifold with tangent bundle $TM$. A vector subbundle
$VM$ of $TM$ is called involutive if the section space $\Gamma(VM)$ of $VM$ is closed under
the Lie bracket ...
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When does the tangent bundle of a manifold admit a flat connection?
Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?
Edit: Formerly, I asked about a flat connection on the frame ...
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Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?
Hello,
I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the ...
- 83
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Transitivity of automorphism group of smooth manifolds
Suppose $M$ is a connected smooth manifold and $x,y \in M$ are two points. Is there always a
diffeomorphism $\phi: M \rightarrow M$ with $\phi(x)= y$ ?
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votes
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answer
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Can a PDE constrain the degree of a $C^\infty$ map germ?
Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold, with $\text{rank}(E)=\text{dim}(M)$. For a section $\sigma$ of $E$ with a zero at $p\in M$, define the degree of the zero at $p$ to ...
- 3,192
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answers
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Proving that the boundary of domain does not contain line segments
Let $D \subset \mathbb{R}^n$ be a bounded domain. If $\partial D$ is a real-analytic sub-manifold, it is not very difficult to show that $\partial D$ does not contain any line segments. Now, assume ...
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2
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Computing self-intersections with complex analysis
It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...
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