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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What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?

H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, (1989), p. xi: ...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic ...
Matthew Niemiro's user avatar
2 votes
0 answers
91 views

If a Compact $n$-Manifold Immerses in $\mathbb{R}^{n+1}$ is there a Locally Flat Immersion?

Suppose that $M$ is a compact, topological $n$-manifold and there is a topological immersion (i.e. local embedding) of $M$ into $\mathbb{R}^{n+1}$. Is there necessarily a locally flat immersion of $M$...
John Samples's user avatar
1 vote
0 answers
237 views

Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle

Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be an oriented $m$-dim ball bundle, i.e. each fiber is an oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\...
DLIN's user avatar
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2 votes
1 answer
123 views

Gluing isotopic smoothings

Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
UVIR's user avatar
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1 vote
0 answers
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Middle Betti number bound of special $4$-manifolds

I was thinking of the following question regarding $4$-manifolds as follows. Let $M$ be a compact, oriented, smooth $4$-manifold with a smooth connected boundary, say $N$. Let $M$ be simply connected ...
piper1967's user avatar
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6 votes
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A particular case of the general converse to the preimage (submanifold) theorem

I was thinking whether it would be possible to develop a converse to the preimage theorem in differential topology and I found the following post: When is a submanifold of $\mathbf R^n$ given by ...
geooranalysis's user avatar
8 votes
0 answers
193 views

A modified version of the converse to the Sard's Theorem

When I learned Sard's Theorem in differential topology by myself, I was thinking whether it would be possible to prove a converse version of the theorem. That is to say, can we somehow show that each (...
pureorapplied's user avatar
1 vote
0 answers
200 views

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 $G$ ...
tota's user avatar
  • 585
6 votes
2 answers
431 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 ...
Ali Taghavi's user avatar
12 votes
1 answer
298 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 ...
Zhenhua Liu's user avatar
16 votes
4 answers
1k 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_{...
Georges Elencwajg's user avatar
16 votes
3 answers
1k 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 ...
Otis Chodosh's user avatar
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1 vote
0 answers
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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\...
James's user avatar
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6 votes
1 answer
513 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 $\mathbb{R}^3$?
Agelos's user avatar
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3 votes
1 answer
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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 ...
Shana's user avatar
  • 237
2 votes
0 answers
116 views

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^*\...
Radeha Longa's user avatar
1 vote
0 answers
108 views

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 ...
Radeha Longa's user avatar
30 votes
1 answer
1k views

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 ...
UVIR's user avatar
  • 971
4 votes
2 answers
979 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 $\pi_1$. ...
Marcell Selltovich's user avatar
2 votes
1 answer
298 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 ...
Akerbeltz's user avatar
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5 votes
1 answer
223 views

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 ...
Yasha's user avatar
  • 469
4 votes
1 answer
240 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 = \...
Matthew Kvalheim's user avatar
3 votes
0 answers
162 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 $\partial A\neq \varnothing$. ...
Random's user avatar
  • 927
9 votes
2 answers
338 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 ...
Arshak Aivazian's user avatar
2 votes
0 answers
81 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 $\...
Bad English's user avatar
6 votes
0 answers
161 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 ...
Arshak Aivazian's user avatar
1 vote
1 answer
134 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-...
Random's user avatar
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0 votes
0 answers
84 views

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 ...
Z. Liu's user avatar
  • 101
1 vote
1 answer
147 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 ...
Steven Landsburg's user avatar
11 votes
1 answer
846 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 ...
Arshak Aivazian's user avatar
2 votes
2 answers
175 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 $f: S^1\times[0,1]\to \mathbb R^3$ ...
Dmitrii Korshunov's user avatar
0 votes
1 answer
334 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 ...
singularity's user avatar
13 votes
2 answers
562 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$ ...
Ceka's user avatar
  • 501
5 votes
1 answer
710 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,...
Overflowian's user avatar
  • 2,523
2 votes
1 answer
236 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 \...
inkievoyd's user avatar
  • 508
2 votes
1 answer
232 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 ...
Ali Taghavi's user avatar
5 votes
0 answers
154 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 ...
maxematician's user avatar
0 votes
0 answers
89 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 $\mathcal{C}^\infty(M,N) \subseteq \mathcal{C}^0(M,N)$, ...
Minkowski's user avatar
  • 571
11 votes
2 answers
554 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 ...
Arshak Aivazian's user avatar
6 votes
0 answers
154 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 ...
UVIR's user avatar
  • 971
5 votes
2 answers
242 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 ...
tota's user avatar
  • 585
9 votes
1 answer
501 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 ...
piper1967's user avatar
  • 1,059
3 votes
2 answers
300 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 ...
Overflowian's user avatar
  • 2,523
4 votes
1 answer
331 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 ...
infinitylord's user avatar
3 votes
0 answers
187 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 $\...
Arnold's user avatar
  • 31
6 votes
0 answers
241 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 ...
tota's user avatar
  • 585
15 votes
2 answers
1k 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 ...
Ilia's user avatar
  • 307
3 votes
0 answers
224 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 ...
Anthony's user avatar
  • 283
5 votes
0 answers
90 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 ...
asv's user avatar
  • 21.1k
9 votes
1 answer
706 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–...
Ya He's user avatar
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