All Questions
145 questions
4
votes
0
answers
177
views
Basis of topology on space of properly embedded smooth manifolds
In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
- 141
25
votes
1
answer
582
views
Does every oriented $3$-dimensional submanifold of $\mathbb{R}^6$ bound an oriented $4$-dimensional submanifold?
In my recent research, I encountered the following problem about embeddings. Let $M^3$ be a closed compact oriented smooth $3$-dimensional submanifold of $\mathbb{R}^6$. Does there exist a compact ...
- 587
6
votes
1
answer
206
views
A stable splitting of linear surjections
Some computations I've been doing in Weiss calculus predict the following stable splitting of the space of linear surjections: $\Sigma^\infty_+ \mathrm{Sur}(\mathbb{R}^n,\mathbb{R}^{m_1+m_2})$
as the ...
- 5,839
5
votes
1
answer
380
views
Proving the Cork Theorem
I am reading Kirby's paper paper "Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds" and I have a question about how to actually prove the cork theorem from the results ...
14
votes
0
answers
326
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
- 3,751
1
vote
0
answers
57
views
extendability of fibre bundles on manifolds with same dimensions
Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where
$M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where
$N'$ is also an $n$-manifold.
Suppose there is fibre ...
- 123
0
votes
1
answer
135
views
Local embedding and disk in domain perturbation
Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
5
votes
1
answer
429
views
Linking number and intersection number
Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
- 125
3
votes
1
answer
200
views
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 ...
- 491
5
votes
0
answers
158
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 ...
- 587
13
votes
1
answer
518
views
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 $2$ ...
- 587
4
votes
0
answers
206
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 ...
- 41
0
votes
1
answer
376
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 ...
- 447
2
votes
0
answers
147
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 ...
- 641
0
votes
0
answers
120
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 ...
- 803
4
votes
1
answer
192
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 ...
- 890
16
votes
0
answers
425
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 ...
- 587
6
votes
0
answers
209
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, ...
- 890
1
vote
0
answers
160
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 $...
- 593
2
votes
0
answers
146
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
13
votes
1
answer
386
views
Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds
Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
- 587
11
votes
1
answer
331
views
Embedded 2-tori in $S^1\times S^4$
I am interested in understanding the smooth isotopy class of embedded 2-tori in $S^1\times S^4$. Is it true that every two homotopic embedded 2-tori in $S^1\times S^4$ are smoothly isotopic? It would ...
- 129
20
votes
2
answers
901
views
Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds
Let $M$ be an $m$-dimensional compact closed smooth manifold and $z\in H_n(M,\mathbb{Z})$ an $n$-dimensional integral homology class, with $m>n.$ Does there exist a pair of $M$ and $z$ so that $z$ ...
- 587
1
vote
1
answer
256
views
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory
As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ...
- 131
8
votes
1
answer
217
views
Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle
Let $M$ be a closed smooth manifold of dimension $n$ and $z\in H_l(M,\mathbb{Z})$ a $k$-dimensional integral homology class. Theorem II.4 of Thom's classical 1954 paper states that for $l< n/2$ or $...
- 587
13
votes
1
answer
459
views
Compact closed aspherical manifolds with vanishing second homology in all the covering spaces
I wonder if there exists a compact closed smooth aspherical manifold $M$ of dimension at least $4$, so that for any covering space $\tilde{M}$ over $M,$ we always have $H_2(\tilde{M},\mathbb{Z})=0$ ...
- 587
3
votes
1
answer
235
views
Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non ...
- 1,097
11
votes
3
answers
1k
views
Computation on characteristic classes
I am organizing a reading seminar on characteristic classes. The audience in the seminar is interested in symplectic and contact manifolds. I work in categorification and would like to compute some ...
- 341
2
votes
0
answers
105
views
Vanishing of Goldman bracket requires simple-closed representative?
Let $\Sigma$ be a connected oriented surface, and $[-,-]\colon \Bbb Z\big[\widehat\pi(\Sigma)\big]\times Z\big[\widehat\pi(\Sigma)\big]\to Z\big[\widehat\pi(\Sigma)\big]$ be the Goldman Bracket. Note ...
- 1,097
2
votes
1
answer
267
views
on second cohomology of $S^1$-manifold
Let $M$ be a closed oriented real manifold with a free smooth circle action. Denote $BS^1$ to be the classifying space of principal circle bundles and $ES^1\rightarrow BS^1$ to be the universal ...
- 113
6
votes
1
answer
414
views
Do $h$-cobordism groups arise from a 'Thom-like' spectrum?
Not thinking about $h$-cobordism, one usually defines a cobordism between manifolds, realizes it is an equivalence relation, chooses an appropriate class of structured manifolds (framed, unoriented, ...
- 609
8
votes
1
answer
429
views
Gluing of orbifolds
Suppose that $P$ and $Q$ are $n$-dimensional orbifolds, with boundaries. Suppose also that there is an isomorphism $f \colon \partial P \rightarrow \partial Q$ (as orbifolds). Is there a way to glue $...
- 781
2
votes
1
answer
130
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 ...
- 803
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 ...
- 803
1
vote
1
answer
147
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-...
- 1,097
6
votes
0
answers
162
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 ...
- 803
8
votes
0
answers
267
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 ...
- 11.9k
2
votes
1
answer
295
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 ...
- 1,097
1
vote
0
answers
284
views
A question on existence of gradient vector field on manifold with boundary
Let $M$ be a compact manifold with smooth boundary $\partial M$. Does $M$ admit a gradient vector field $\nabla u$, which has no zeros, i.e. $\nabla u(x)\neq 0$, $\forall x\in M\cup\partial M$?
Thanks ...
- 51
6
votes
1
answer
222
views
Stable smoothing of topological manifolds relative to an embedding
Let $M$ be a topological manifold. We know that $M$ is stably smoothable if and only its tangent microbundle, up to stabilization, admits a reduction to vector bundle.
Now I wonder if there is a ...
- 803
17
votes
1
answer
898
views
Does there exist a closed manifold with vanishing reduced rational cohomology but nonvanishing odd torsion cohomology?
Question: Let $p$ be an odd prime. Does there exist a closed manifold $M$ with $\widetilde H^\ast(M; \mathbb Q) = 0$ but $\widetilde H^\ast(M; \mathbb F_p) \neq 0$?
When $p = 2$, an example is given ...
- 63.9k
0
votes
0
answers
266
views
Define a characteristic class on a simplicial complex (non-manifold)
Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed ...
- 10.5k
9
votes
1
answer
588
views
Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)
The classification of oriented compact smooth manifolds up to oriented cobordism is one
of the landmarks of 20th century topology. The techniques used there form the part of the foundations of ...
- 10.5k
11
votes
0
answers
335
views
Isotopy on embedded 3-manifolds in 4-manifolds
Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$...
- 3,751
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 10.6k
0
votes
1
answer
154
views
Why does $X_0\times S^1\simeq X-X_0$? [closed]
Let $X$ be an $n$-dimensional connected smooth manifold, and let $X_0$ be an embedded $(n-2)$-dimensional compact submanifold of $X$ with the trivial normal bundle. How do we get inclusion?
$$X_0\...
- 563
0
votes
1
answer
284
views
Künneth formula and induced map in homologies
Let $X,Y,Z$ be smooth connected manifolds and $f \colon X \times Y \rightarrow Z$ a smooth map. Suppose that we have $H_{*}(X \times Y; \mathbb{Z})$ is isomorphic to $\bigoplus_{p+q=*}(H_{p}(X; \...
- 369
8
votes
0
answers
222
views
Representing the fundamental class of an aspherical manifold in the bar complex
Suppose $M$ is a compact orientable aspherical manifold and $G$ its fundamental group. Is there a nice description of representatives of the fundamental class of $M$ and its dual in the (homogenous) ...
- 1,174
1
vote
0
answers
102
views
DA structure of a Dehn twist
I am trying to find the DA bordered homology structure of a Dehn twist.In https://arxiv.org/pdf/0810.0687v6.pdf page 255 bottom the authors tabulate the differentials of the right module(A side) of ...
- 99
8
votes
0
answers
219
views
Computation of the 3-dimensional $\mathbb{Z}/m$-equivariant spin cobordism group (with possibly non-empty fixed-point set)?
$\newcommand{\odd}{\mathrm{odd}}\newcommand{\ev}{\mathrm{ev}}$Consider tuples of the form $(Y,\mathfrak{s},\widehat{\sigma})$ where: $Y$ is a closed oriented 3-manifold, $\mathfrak{s}$ is a spin ...
- 299