All Questions
151 questions
8
votes
0
answers
241
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+300
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 501
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
7
votes
1
answer
202
views
Lipschitz bounds and homotopy groups of diffeomorphism groups
Let $M$ denote a closed Riemannian manifold. Let $\mathrm{Diff}_0^L(M)$ denote the supspace of the identity component of the diffeomorphism group $\mathrm{Diff}_0(M)$ of diffeomorphisms with Lipschitz ...
- 1,174
6
votes
0
answers
129
views
Are there isospectrally equivalent exotic spheres?
Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
I would be happy ...
0
votes
1
answer
155
views
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 ...
- 2,792
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
6
votes
2
answers
402
views
"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, ...
- 447
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
208
views
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 ...
- 501
15
votes
6
answers
2k
views
Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
I'm now attending a reading seminar on the algebraic topology.
The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).
In those ...
- 1,013
1
vote
0
answers
153
views
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 ...
- 111
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
1
vote
0
answers
151
views
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 ...
- 447
0
votes
0
answers
85
views
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}...
29
votes
4
answers
3k
views
Conceptual proof of classification of surfaces?
Every compact surface is diffeomorphic to $S^2$, $\underbrace{T^2\#\ldots \#T^2}_n$, or $\underbrace{RP^2\#\ldots \#RP^2}_n$ for some $n\ge 1$.
Is there a conceptual proof of this classification ...
- 43.2k
10
votes
1
answer
570
views
Is every retraction homotopic to a smooth retraction?
I am not an expert in Differential Topology, so let me apologize if this question admits a straightforward answer. I checked some standard references, but I could not find one.
Let $M$ be a smooth $n$-...
- 66.3k
13
votes
4
answers
2k
views
Fundamental groups of compact Kähler manifolds
This is a sort of a follow-up to this question, and especially to Sean Lawton's answer: The book Fundamental Groups of compact Kähler manifolds (which, in my opinion, is one of the best mathematics ...
- 96.4k
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
31
votes
1
answer
1k
views
What results about the topology of manifolds depend on the dimension mod 3?
There are a lot of interesting results about the topology of manifolds that depend on the dimension of the manifold mod 2, mod 4, or mod 8. The simplest ones involve the cup product
$$ \smile \colon ...
- 22.3k
0
votes
0
answers
58
views
Role of basins of attraction in the Morse decomposition
Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by
$$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
- 247
19
votes
1
answer
989
views
Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
Suppose we have two three-...
- 293
10
votes
2
answers
2k
views
Parallelizability of 3-manifolds
Robert Bryant's answer here ( https://mathoverflow.net/a/149496/85500 ) states that any orientable 3-manifold is parallelizable.
Previously I was under the impression that only closed (compact & ...
- 2,792
5
votes
2
answers
249
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 ...
- 585
-2
votes
1
answer
189
views
Topologies in the vicinity of Euclidean space
Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$.
Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ ...
- 521
14
votes
1
answer
681
views
When does an open manifold admit two linearly independent vector fields?
$\DeclareMathOperator{\span}{span}$
$\DeclareMathOperator{\co}{H}$
$\newcommand{\kk}{\mathbb{F}}$
$\newcommand{\qq}{\mathbb{Q}}$
$\newcommand{\zz}{\mathbb{Z}}$
$\newcommand{\rr}{\mathbb{R}}$
$\...
- 1,726
15
votes
3
answers
2k
views
Examples of odd-dimensional manifolds that do not admit contact structure
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
- 2,533
5
votes
0
answers
121
views
How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?
I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-...
- 9,019
5
votes
0
answers
133
views
On the vertical cohomology of a fibered manifold
Let $\pi:Y\rightarrow X$ be a $C^\infty$ fibered manifold (all constructions, unless otherwise stated are over the smooth manifold category) with $\Omega_k$ the sheaf of (smooth) $k$-forms on $Y$.
...
- 2,792
19
votes
8
answers
2k
views
Theorems that led to very successful research programs in Geometry and Topology [closed]
In the recent times I have heard a lot about the following:
The Atiyah-Singer Index theorem
H-principle of Gromov ( and others )
It seems to me that these results led to decades of successful ...
8
votes
1
answer
426
views
Orbifolds are Thom-Mather stratified spaces
Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space?
edit: after some search, I found the proof should be contained in either
GIBSON, C....
- 803
1
vote
0
answers
246
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:\...
- 1,915
18
votes
1
answer
1k
views
Approximation of homeomorphism by diffeomorphism
Let $M$ be a smooth closed manifold. Let $f\colon M\to M$ be a homeomorphism.
Does there exist a sequence of diffeomorphisms $f_i\colon M\to M$ which conveges to $f$ uniformly, i.e. in $C^0$-...
- 21.8k
1
vote
0
answers
139
views
Cartesian product of spin manifolds [closed]
Is it true that the Cartesian product of two spin manifolds is spin?
5
votes
1
answer
408
views
A question related to fiber bundle
Let $f:\mathbb{C}^3 \to \mathbb{C}$ be a morphism of varieties such that it is a smooth fiber bundle. Can I say that the fiber is $\mathbb{C}^2$?
- 1,177
6
votes
0
answers
297
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 ...
- 585
4
votes
1
answer
304
views
Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for $n=2k+1$
Atiyah, Bott, and Shapiro paper on Clifford Modules around page 10 shows two facts.
1 - There is a lift $U(n) \to Spin^c(2n)$ from $U(n) \to SO(2n)\times U(1)$. Also an embedding (injective group ...
- 317
9
votes
0
answers
338
views
Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?
This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case.
Let $M$ be a ...
- 1,726
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
11
votes
1
answer
379
views
Smooth structure on direct product
Let $M$ be the $E_8$ manifold. Is there a closed manifold $N$ such that $M\times N$ is smoothable? What is the smallest possible dimension of $N$?
user164740
2
votes
0
answers
137
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^*\...
- 563
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
254
views
$SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]
The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal ...
- 317
13
votes
2
answers
1k
views
Realizing cohomology classes by submanifolds
In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. ...
- 5,824
16
votes
1
answer
1k
views
Is there an area-preserving diffeomorphism of the disk which is nowhere conformal?
This question is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk.
Does there exist a smooth area-preserving diffeomorphism $f:D \to D$ that does not have conformal points?
I ...
- 6,741
2
votes
0
answers
168
views
Geometric sets determined by chains (for integration and Stokes' theorem)
I have asked a similar question on mathSE more than a year ago, which received no answers, only a few comments which did not really help me. I am now re-asking this question here but reformulated ...
- 2,792
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
17
votes
1
answer
1k
views
Direct proof that Chern-Weil theory yields integral classes
Suppose $E$ is a complex vector bundle of rank $n$ on a compact oriented manifold (both assumed smooth). Let $h$ be a Hermitian metric on $E$, and let $A$ be a Hermitian connection on $E$ and $F_A$ ...
- 1,761
8
votes
1
answer
228
views
Isomorphisms of Pin groups
My goal is to identify the $Pin$ group
$$
1 \to Spin(n) \to Pin^{\pm}(n) \to \mathbb{Z}_2 \to 1
$$
such that $Pin^{\pm}(n)$ are isomorphisms to other more familiar groups.
My trick is that to look at ...
- 10.5k
12
votes
2
answers
597
views
Steenrod powers of Pontryagin classes
It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true ...
- 1,452
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