All Questions
151 questions
64
votes
1
answer
4k
views
A dictionary of Characteristic classes and obstructions
I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
In an effort to ...
- 7,789
63
votes
0
answers
2k
views
Are there periodicity phenomena in manifold topology with odd period?
The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...
- 118k
32
votes
2
answers
2k
views
Converse to Stokes' Theorem
Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : \Lambda^...
- 64k
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
29
votes
2
answers
2k
views
A simple proof that parallelizable oriented closed manifolds are oriented boundaries?
So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
- 7,609
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
26
votes
2
answers
2k
views
Euler characteristic and universal cover
Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$.
My question is: does this imply that $\chi(M)=0$?
This is clear if ...
- 1,452
24
votes
1
answer
1k
views
All fiber bundles over $S^2$ extendable to $\mathbb{C}P^\infty$?
I ran into the following sanity check. Is the following statement true?
Every smooth fiber bundle (with compact fiber) over $S^2$ can be extended to a smooth fiber bundle over $\mathbb{C}P^\infty$ (...
- 707
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 ...
19
votes
3
answers
3k
views
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 ...
- 8,512
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
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
18
votes
3
answers
922
views
When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$
For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$
In my case ...
- 1,893
18
votes
1
answer
1k
views
Is the restriction map for embeddings of manifolds with boundary a fibration?
Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...
- 2,974
17
votes
2
answers
737
views
Are the associative grassmannian and the quaternionic projective plane diffeomorphic?
I have a doubt which I hope the MO community can quickly resolve.
The associative grassmannian is an eight-dimensional homogeneous space $G_2/SO(4)$. It can be identified with the space of ...
- 33.3k
17
votes
5
answers
3k
views
Homologically trivial submanifolds
Unuseful prequel
Let $M$ be a (compact, oriented, differentiable) manifold. Before knowing anything about homology theory a naif but clever mathematician may want to measure the holes in $M$ by the ...
- 14.7k
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 ...
- 64k
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
16
votes
2
answers
1k
views
Classification of $O(2)$-bundles in terms of characteristic classes
I had asked this question in stackexchange but there seems to be no consensus in the answer
It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by ...
- 2,816
16
votes
2
answers
605
views
What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?
The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...
- 64k
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
16
votes
0
answers
426
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
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
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
15
votes
3
answers
2k
views
Which submanifolds are zero sets of $\mathbb{R}^n$-valued maps?
If $M$ is a smooth, compact, orientable manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal ...
- 972
15
votes
1
answer
1k
views
Higher Cerf Theory
Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that $$f(x_1,\dots,x_n)=-x_1^2-x_2^2-\dots-...
- 865
15
votes
0
answers
1k
views
Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle
My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
- 648
14
votes
2
answers
1k
views
Converse of Poincaré-Hopf theorem
Let $M$ be a connected, compact, oriented manifold of dimension $n<7$. If any two maps $M \to M$ having equal degrees are homotopic, must $M$ be diffeomorphic to the $n$-sphere?
- 471
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
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
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
13
votes
1
answer
637
views
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 ...
- 3,212
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
12
votes
1
answer
840
views
Reference request: Topology on the space of smooth compact submanifolds
In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...
- 121
11
votes
1
answer
724
views
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 ...
- 1,415
11
votes
2
answers
988
views
first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles
Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to $SO(...
- 2,223
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
11
votes
1
answer
940
views
Equivariant sections of fiber bundles
One of the fundamental facts in fiber bundle theory is the following result for existence and extension of sections (see Thm. 9 in this paper of Palais, and compare with Thm. 12.2 in Steenrod's book):...
- 7,194
11
votes
1
answer
562
views
Relation between Morse Theory and integration against Euler Characteristic
I'm studying Robert Ghrist papers on integration against Euler Characteristic. I am particularly interested in the relation with Morse Theory. I am trying to understand the proof of Theorem 25.1 (page ...
- 909
11
votes
0
answers
650
views
Triangulation of manifolds with corners
Let's begin with some definitions:
A (smooth) manifold with corners is a Hausdroff (and second countable if you want) space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \...
- 909
10
votes
2
answers
1k
views
Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated?
Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of ...
- 977
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 & ...
- 3,292
10
votes
1
answer
386
views
A symmetric embedding of manifolds
Assume that $M$ is a manifold.
Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots x_{i},\...
- 356
10
votes
2
answers
730
views
Representability of the sum of homology classes
This is probably a very simple question, but I have not found it addressed in the references that I know. Let $M$ be a closed and connected orientable $d$-dimensional manifold ($d\leq 8$) and let $[\...
- 2,816
10
votes
1
answer
732
views
What's the sufficient or necessary conditions for a manifold to have Lie group structure?
For example, given a Lie group, its fundamental group must be Abelian. So $\Sigma_g$ ($g>1$) can't have Lie group structure. We also know for $S^n$ only $n=0,1,3$ can have Lie group structures.
...
- 249
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
10
votes
2
answers
876
views
Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?
The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...
- 888
10
votes
1
answer
1k
views
Classification of oriented vector bundles of rank 5 over closed oriented 5-manifolds
I am looking for a complete classification in terms of characteristic classes and "computable" (preferably geometric) invariants. There is this work where the authors classify oriented vector bundles ...
- 2,830
10
votes
0
answers
281
views
Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$
Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...
- 28.9k
9
votes
1
answer
384
views
embedding of quaternionic projective spaces
Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any ...
- 1,990