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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Question regarding affine fibre bundles

Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...
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6 votes
2 answers
369 views

The convex hull of a manifold whose cobordism class is trivial

Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class. Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...
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12 votes
1 answer
177 views

Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
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15 votes
4 answers
975 views

Can one glue De Rham cohomology classes on a differential manifolds?

Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k_{...
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15 votes
3 answers
919 views

Converse to Hopf degree theorem

Below, I mean smooth oriented closed connected manifolds and smooth maps (but am happy to hear about the topological category, or unoriented manifolds, etc instead). Say that $X^n$ has the Hopf ...
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1 vote
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Results on compact slices in a regular foliation

Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...
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  • 111
6 votes
1 answer
267 views

Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$?

Let $M$ be a simply connected, (orientable), non-compact, 3-manifold without boundary. Must $M$ be homeomorphic with a topological subspace of $\mathbb{R}^3$?
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  • 955
3 votes
1 answer
286 views

How to show that $\text{Man}(M,\mathbb{R}^n)\cong \mathbb{R}\text{-Alg}(C^\infty(\mathbb{R}^n),C^\infty(M))$?

I'm trying to show that manifolds are affine, i.e. $\text{Man}(M,N)\cong \mathbb{R}\text{-Alg}(C^\infty(N),C^\infty(M)) $. If I could show this for $N=\mathbb{R}^n$, then I know how to do the rest ...
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  • 203
2 votes
0 answers
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Question about spin map

I'm confused with the following definition of a spin map. A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...
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1 vote
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Existence of a local spinor bundle

I am confused about the existence of a local spinor bundle. My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...
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29 votes
1 answer
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Are homeomorphic representations isomorphic?

Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are ...
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  • 751
3 votes
2 answers
320 views

Representation of fundamental group and flat connections

I read Differential Geometry Of Complex Vector Bundles by Kobayashi, and he says there that a vector bundle $E$ has flat connection is equivalent to $E$ being defined by a representation of $\pi_1$. ...
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2 votes
1 answer
209 views

Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely

Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...
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4 votes
1 answer
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Is the smooth singular simplicial set of a smooth manifold a Kan complex?

It is classical that the singular simplicial set of a topological space is a Kan complex. This is elementary and already due to presumably Kan. Q: Is the smooth singular simplicial set of a smooth ...
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4 votes
1 answer
188 views

A cobordism theory from Hirsch's "Differential Topology" (reference request)

The following is exercise 5 on p. 176 in Hirsch's "Differential Topology" (corrected 6th printing): Let $\eta = (p,E,B)$ be a fixed vector bundle over a compact manifold $B$, $\partial B = \...
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3 votes
0 answers
47 views

The boundary of the transversal pre-image of a submanifold with boundary

A similar question on MSE without answer. Let $M, N$ be smooth manifolds such that $\partial N=\varnothing$. Let $A$ be a smoothly embedded submanifold of $N$ such that $\partial A\neq \varnothing$. ...
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8 votes
2 answers
233 views

Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds

This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following ...
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2 votes
0 answers
61 views

Gysin homomorphism of an inclusion to Kähler tubular neighborhood

Let $Z\subset U$ be a Kähler tubular neighborhood of a compact manifold $Z$ of codimension $r$. Consider de Rham complexes of smooth differential forms $\Lambda^{*,*}(Z),\Lambda^{*,*}(U)$, let $\...
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4 votes
0 answers
109 views

Is the category of diffeological spaces a full subcategory of locally ringed spaces?

It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here). Is a similar ...
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1 vote
1 answer
88 views

Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood

A similar post on MSE without answer. Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold of $M$ of co-...
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The closure of the subgroup generated by a vector field may not be compact

Suppose $X$ is a vector field on a manifold $M$, consider the one parameter group: $$L=\left\{\phi^t_X: t\in\mathbb{R}\right\}$$ where $\phi^t_X$ is the flow of the vector field $X$, which sends $p\in ...
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2 votes
1 answer
125 views

Uniqueness of "stretching" (subject to constraints) for a two-dimensional figure

The New York Times, reporting on Dennis Sullivan's Abel prize, recounts the incident that lured Sullivan from chemical engineering to mathematics: One day during an advanced calculus lecture, the ...
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7 votes
1 answer
429 views

Intuition/meaning behind/physical content of the concept of a smooth structure

Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological ...
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2 votes
2 answers
156 views

An equivalence relation on knots similar to concordance

Let $L_1$ and $L_2$ be two nonintersecting picewise-linear or smooth knots in $\mathbb R^3$. Suppose they are ambient isotopic. Does there exist an embedded surface $f: S^1\times[0,1]\to \mathbb R^3$ ...
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  • 1,214
0 votes
1 answer
210 views

Sheaf of Kähler differentials for complex manifold

Let $X \subset \mathbb{C}^n$ be an analytic set which means that it is the zero locus of holomorphic functions $f_1,f_2,\dotsc,f_n$ on $\mathbb{C}^n$ and suppose that there is a singular locus ...
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9 votes
1 answer
384 views

When are bundles of odd and even differential forms isomorphic?

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ ...
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  • 389
5 votes
1 answer
322 views

Kirby diagrams: sliding 1-handles over 1-handles and ribbon disks

Consider the Kirby diagram $ D$ given by a 2-component unlink, both dotted circles. In general, when performing a 1-handle slide over another 1-handle, the band chosen must not link any dotted circle,...
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2 votes
1 answer
114 views

Quantitative results for stabilizing tangent bundles of homology spheres

I'll begin with a broad question: if $M$ is a smooth manifold and $E \to M$ is a stably trivial bundle, can one determine lower bounds on the rank $k$ of the trivial bundle needed such that $E \oplus \...
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2 votes
1 answer
166 views

Equidistant points on a compact Riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows: $K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at ...
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5 votes
0 answers
129 views

Potential theory as a tool in extrinsic flows

Let $M \subseteq \mathbb{R}^n$ be a submanifold. For a point $x$ disjoint from $M$, we can define the electric potential $\Phi(x) = \int_M \frac{dM}{|x-m|^{n-2}}$, which is smooth and harmonic where ...
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0 votes
0 answers
75 views

Topologies in $\mathcal{C}^\infty(M,N)$

Naively, one could topologise the set of smooth (ie $\mathcal{C}^\infty$) maps between two smooth manifolds $M \to N$ with the subspace topology $\mathcal{C}^\infty(M,N) \subseteq \mathcal{C}^0(M,N)$, ...
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10 votes
2 answers
413 views

Homotopy properties of Lie groups

Let $G$ be a real connected Lie group. I am interested in its special homotopy properties, which distinguish it from other smooth manifolds For example $G$ is homotopy equivalent to a smooth compact ...
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6 votes
0 answers
128 views

Uniqueness of normal microbundle of a smooth embedding

Suppose $M$ is a topological manifold and $\iota: N\hookrightarrow M$ be a submanifold. A normal microbundle of $N$ consists of an open neighborhood $U$ of $N$ and a retraction $\pi: U \to N$ such ...
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  • 751
4 votes
1 answer
145 views

Patching up two trivial fibre bundles induces homology equivalence

I was wondering to ask this question may be it's a silly one. I could not prove or disprove it. Let $X,Y$ be smooth connected manifolds. Let $X=X_1\cup X_2$ ($X_i$'s sub-manifold of $X$) and $X_1 \cap ...
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  • 155
6 votes
0 answers
131 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 ...
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  • 851
3 votes
2 answers
281 views

Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$

$\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$. I would like ...
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4 votes
1 answer
241 views

Poincaré–Bendixson Theorem on a compact, connected, orientable, two-dimensional manifold

I'm currently reading the article "A Generalization of a Poincaré–Bendixson Theorem to Closed Two-Dimensional Manifolds" by Arthur Shwartz. The paper first establishes a result for minimal ...
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2 votes
0 answers
104 views

Proper isotopy of proper embeddings of manifolds

We know from Theorem 2.2 in(http://www.map.mpim-bonn.mpg.de/Embeddings_of_manifolds_with_boundary:_classification#6.2) that Any two smooth embeddings of closed oriented $n$-manifold(n>1) in $\...
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5 votes
0 answers
100 views

Regarding homology of fiber bundle

Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...
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  • 155
15 votes
2 answers
873 views

Diffeomorphism group of the projective plane

First of all, I am interested in the general case of a non-orientable manifold but let's for now consider the projective plane $\mathbb{R}P^2.$ In short, I am curious if there is any relation between ...
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3 votes
0 answers
167 views

Standard sutured (?) Heegaard splitting

I am trying to make sense of what is going on in [Cas16] in terms of diagrams. Let me sum up the construction a bit, where $n\leqslant k$ are integers and $b\geqslant 1$ as well. $C_{k,b,n}$ denotes ...
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4 votes
0 answers
81 views

Equivariant imbedding of compact manifold

Let $G$ be a compact Lie group smoothly acting on a smooth compact manifold $X$. Is it true that there exists a smooth $G$-equivariant imbedding of $X$ into a Euclidean space acted linearly (and ...
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  • 19k
7 votes
1 answer
382 views

Is there a version of the Poincaré–Hopf theorem for manifold with corners?

As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...
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3 votes
0 answers
68 views

Vietoris-Begle type result for differentiable fiber bundle

In Vietoris-Begle Theorem, we consider a closed and surjective map between two paracompact and Hausdorff spaces and we get some relation involving the homologies of the fiber, total space, and the ...
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  • 851
1 vote
0 answers
81 views

Modular cycles?

It is well known that cocycles (differential forms) and cycles share many properties through duality (e.g., de Rham). I've been reading about modular forms recently and I came with a very naive ...
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-1 votes
1 answer
182 views

Essential simple closed curves in a torus [closed]

Definition: By a closed curve in a surface $S$ we will mean a continuous map $S^1 \to S$. We will usually identify a closed curve with its image in $S$. A closed curve is called essential if it is not ...
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1 vote
1 answer
146 views

Local discriminant variety

I'm looking for good (as simple as it is possible) reference for the local discriminant variety. I need it in the following situation: I have an unfolding $F: (\mathbb{K}^n \times \mathbb{K}^p, 0) \to ...
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8 votes
1 answer
353 views

Regarding the surgery construction in "A procedure for killing homotopy groups of differentiable manifolds" by Milnor

In the first section of "A procedure for killing homotopy groups of differentiable manifolds", Milnor gives the surgery construction as follows. Let $W$ be an $n=p+q+1$ dimensional manifold. ...
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7 votes
0 answers
237 views

$\mathbb RP^n$ bundles over the circle, II

EDIT: I fixed the issue pointed out by Nicholas Tholozan, thanks for sheding light on this! This question is written as a follow-up to this one. Both answers there are great, but my impression is ...
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2 votes
1 answer
246 views

In which dimensions is it true that every topological ball embedded by a smoothly embedded sphere is a smoothly embedded ball?

I asked a question on MSE with no answer. Here is my question in the generalized version. Question 1: Suppose we are given a connected three-manifold $M$ (possibly non-compact, or non-orientable) and ...
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