Questions tagged [smooth-manifolds]
Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].
1,236
questions
14
votes
0
answers
464
views
Which spherical space forms embed in $S^4$?
Is there any hope of getting a classification of which 3-dimensional spherical space forms are smoothly embeddable in $S^4$? I read that lens spaces cannot embed in $S^4$, but some other spherical ...
- 161
14
votes
0
answers
545
views
Reference for a proof of the fiberwise Stokes theorem
The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary,
the difference between the fiberwise integral of the differential and the ...
- 36.3k
13
votes
2
answers
2k
views
Can a simply connected manifold satisfy $𝑀\simeq 𝑀\times 𝑀$?
Let $M$ be a simply connected, (finite dimensional) smooth manifold. Is it possible that $M$ is homotopy equivalent to $M\times M,$ without $M$ being contractible? This would imply $\pi_n(M)\times\...
- 1,188
13
votes
2
answers
2k
views
Spectrum of Ring of Smooth Functions on $\mathbb{R}^n$
When we define smooth manifold, we starting with topological space $M$ which localy homeomorphic to $\mathbb{R}^n$ and setting up sheaf $\mathscr{F}(M)$ of functions on it which localy isomorphic to $\...
- 171
13
votes
4
answers
2k
views
The Schwartz Space on a Manifold
I asked this question a couple of days ago on math.stackexchange, but have yet to receive a response, so I have decided to post this here.
This question is also vaguely related (both questions arose ...
- 1,220
13
votes
3
answers
1k
views
How to disjoint two cycles with zero intersection?
Suppose that $M^n$ is a smooth connected orientable manifold and $Z^k$ with $Z^{n-k}$ are two real cycles in $M^n$ with zero index of intersection $Z^k\cdot Z^{n-k}=0$ (these cycles are submanifolds ...
- 14k
13
votes
2
answers
719
views
Can every element of a homotopy group of a smooth manifold be represented by an immersion?
I originally posted this on MSE but didn't get much of a response, so I'll attempt to post it here. Let me know if this is not appropriate.
Let $M$ be a smooth manifold of dimension $n$. Let $\alpha \...
- 1,735
13
votes
3
answers
2k
views
What is the characteristic property of surjective submersions?
In Lee's Introduction to smooth manifolds he states that given smooth manifolds $X,Y$ and a surjective submersion, $f:X \rightarrow Y$, then $f$ is a smoothly final map, that is for any further smooth ...
- 2,043
13
votes
1
answer
533
views
Identifying two definitions of orientation on a vector space
Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:
A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
- 133
13
votes
2
answers
689
views
Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?
It's easy to derive a presentation of the fundamental group of a 4-manifold if you have a Kirby diagram: The 1-handles are generators and the 2-handles are the relations. The 3- and 4-handles, which ...
- 5,545
13
votes
1
answer
584
views
Are there examples of Einstein manifolds with unbounded curvature?
Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$
I'm looking for these examples because they might provide a counter-example to a ...
- 649
13
votes
0
answers
174
views
Is there a Handle Approximation theorem?
The cellular approximation theorem states that given a continuous map between two CW complexes $f : X \to Y$, then $f$ is homotopic to a cellular map - that is some map $f'$ with $f'(X_n) \subset Y_n$ ...
- 5,319
12
votes
3
answers
839
views
$A_{\infty}$-structure on closed manifold
Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...
- 1,974
12
votes
2
answers
3k
views
Stokes' theorem etc., for non-Hausdorff manifolds
This question is prompted by another one.
I want to motivate the definition of a scheme for people who know about manifolds(smooth, or complex analytic). So I define a manifold in the following way.
...
- 7,272
12
votes
3
answers
2k
views
In Diff, are the surjective submersions precisely the local-section-admitting maps?
Question as in title (Diff = category of smooth manifolds and smooth maps)
I thought I'd convinced myself this is true, so this is just a sanity check.
Also, what about for settings other than ...
- 33.9k
12
votes
5
answers
1k
views
Which manifolds admit a diffeomorphism of order $n$?
Let $n>1$. Which smooth manifolds admit a diffeomorphism $f$ of order $n$?
For a closed orientable surface $S_g$ of genus $g$ and $n=2$ the answer is in the affirmative, since $S_g$ can be ...
- 637
12
votes
1
answer
804
views
Manifolds with nonwhere vanishing closed one forms
I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $N\times S^1$.
- 707
12
votes
3
answers
1k
views
Is the class of n-dimensional manifolds essentially small?
Question: Consider the proper class of all $n$-dimensional smooth manifolds. If we take the equivalence classes where two manifolds are identified if there exists a diffeomorphism between them, is ...
- 37.6k
12
votes
3
answers
953
views
Fixed point set of smooth circle action
Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even ...
- 11.9k
12
votes
4
answers
2k
views
Book on manifolds from a sheaf-theoretic/locally ringed space PoV
I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds ...
- 1,054
12
votes
3
answers
3k
views
Symmetric and anti-symmetric parts of the covariant derivative of a connection
The following is an excerpt from Sharpe's Differential Geometry - Cartan's Generalization of Klein's Erlangen Program.
Now we come to the question of higher derivatives. As usual in modern
...
- 10.3k
12
votes
2
answers
762
views
Topological obstructions to existence of immersion
Let $M$ be a smooth, non-compact manifold.
a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ?
b) If not, are there some concrete ...
- 1,225
12
votes
7
answers
1k
views
Cohomology classes annihilated by pullbacks
A friend of mine is interested in examples of the following situation:
an oriented smooth fiber bundle $\pi \colon M \to B$ with $M$ and $B$ compact
and a non-zero class $a \in H^3(B; \mathbb{Q})$
...
- 4,696
12
votes
1
answer
1k
views
Intuitive Aproach to Dolbeault Cohomology [closed]
(Duplicated from math.stackexchange)
I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. ...
- 2,071
12
votes
1
answer
703
views
Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant
I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.
All manifolds are closed, smooth and have dimensions $n\ge 5$.
The Atiyah-Shapiro-Bott-Orientation gives ...
- 2,954
12
votes
2
answers
506
views
Homeo-Fixed point property
Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
...
- 312
12
votes
2
answers
550
views
Self homeomorphisms of $S^2\times S^2$
Every matrix $A\in SL_2(\mathbb{Z})$ induces a self homeomorphism of $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. For different matrices these homeomorphisms are not homotopic, as the induced map on $\...
- 10.3k
12
votes
2
answers
2k
views
What is the infinite-dimensional-manifold structure on the space of smooth paths mod thin homotopy?
This question is motivated by the recent paper An invitation to higher gauge theory by Baez and Huerta, and the 2007 paper Parallel Transport and Functors by Schreiber and Waldorf.
Let $M$ be a ...
- 53.1k
12
votes
3
answers
3k
views
Non-Lie Subgroups
A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal $G$-bundle is a Lie subgroup of $G$ (Cartan had earlier asserted this, but apparently without proof).
...
- 2,706
12
votes
1
answer
301
views
Approximate classifying space by boundaryless manifolds?
As pointed out by Achim Krause, any finite CW complex is homotopy equivalent to a manifold with boundary (by embedding into $\mathbb R^n$
and thickening), and so every finite type CW complex can be ...
- 123
12
votes
2
answers
3k
views
Derivative of the flow for ODEs on manifolds
Let $\mathbf V \colon [0,T] \times \mathbb R^d \to \mathbb R^d$ (for $T>0$) be a given, bounded smooth vector field and let $\mathbf X=\mathbf X(t,x)$ be its flow, i.e. the unique solution to the ...
- 588
12
votes
1
answer
389
views
Are algebras of smooth functions formally smooth?
Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$?
If it helps, feel free to assume that $M$ is compact.
(This is not a joke ...
- 5,795
12
votes
2
answers
353
views
Is $C^r(M)$ non-isomorphic to $C^s(N)$ for $r\neq s$ and nontrivial manifolds $M,N$?
This is an obvious continuation of an MO question. Let $r,s\in\mathbb N\cup\{\infty\}$ with $r\neq s$, and $M,N$ two connected manifolds of positive dimension (which roots out the trivial case of a ...
- 2,003
12
votes
1
answer
252
views
Homotopies with prescribed regular values
Let $M_1$ and $M_2$ be connected smooth manifolds and let $f_0,f_1:M_1 \rightarrow M_2$ be homotopic smooth maps such that some fixed point $p \in M_2$ is a regular value for both $f_0$ and $f_1$. ...
- 321
12
votes
0
answers
255
views
Smooth dual cell structure
Let us consider a closed oriented smooth manifold M. It is well known that a smooth combinatorial triangulation can be constructed for it. That is to say, a homeomorphism from the geometric ...
- 121
12
votes
0
answers
403
views
The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the ...
- 7,549
12
votes
0
answers
377
views
Two ways a manifold can have little symmetry
Let $M$ be a closed connected smooth oriented manifold. The following two properties - that $M$ can either enjoy or not - intuitively both mean that $M$ has very little symmetry:
(a) Every self-map $...
- 11.9k
12
votes
0
answers
479
views
Exotic smoothness and Parallelizability
Regarding the parallelizability of the Milnor's seven dimensional exotic spheres:
Parallelizability of the Milnor's exotic spheres in dimension 7
The following question naturally arises:
Suppose ...
- 1,226
11
votes
3
answers
2k
views
Is each closed convex set a manifold with corners?
Assume that $C$ is a convex set in $\mathbb{R}^{n}$ with non empty interior.
Then consider its closure, is it a smooth manifold with corners?
Edit:
1) The closure of $C$ should be a smooth manifold ...
- 1,424
11
votes
5
answers
2k
views
Ricci Curvature in infinite dimensions?
Is there a good notion of "Ricci curvature" in infinite dimensions?
My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...
- 235
11
votes
1
answer
843
views
Critical dimensions D for "smooth manifolds iff triangulable manifolds"
I am aware that at least for lower dimensions,
"smooth manifolds iff triangulable manifolds"
at least for dimensions below a certain critical dimensions D.
My question is that for
For ...
- 10.3k
11
votes
2
answers
306
views
Triangulation with simplices of same volume
Let $M$ be a Riemannian smooth compact manifold.
It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ?
...
11
votes
1
answer
2k
views
Finite-dimensionality for de Rham cohomology
I was browsing through the litterature, hoping to find sufficient and necessary conditions for a smooth manifold to have finite-dimensional de Rham cohomology, but I can't find any satisfactory answer....
- 397
11
votes
2
answers
714
views
The ring $C^{\infty}(M)$?
Let $M$ be a smooth paracompact manifold. I think that the ring $C^{\infty}(M)$ contains many (possibly almost all?) geometric or topological information about $M$.
(e.g. Let $E$ be a vector bundle ...
- 678
11
votes
1
answer
381
views
Twisting bordism classes
Let $X$ be a reasonable topological space (I'd be happy to assume that $X$ is a smooth closed manifold) and let $f\colon M^n \rightarrow X$ be a continuous map from a smooth oriented $n$-manifold $M^n$...
- 113
11
votes
3
answers
957
views
Waldhausen $K$-theory for $G$-spaces
I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it?
Let $G$ be a finite group. We know that ...
- 54.4k
11
votes
1
answer
877
views
Smooth four-manifolds with contractible universal cover
Let $X$ be a smooth compact four-manifold with definite non-trivial intersection form. Can the universal cover of $X$ be contractible?
It semms to me that the answer is negative when $X$ is simply ...
- 123
11
votes
1
answer
2k
views
Does pullback in the category of smooth manifolds always exists?
I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist.
Remarks:
1) A pullback in a certain category is defined as ...
- 6,621
11
votes
1
answer
324
views
Does the Lie algebra of vector fields $\mathfrak{X}(M)$ determine the diffeomorphism class of a manifold $M$?
Let $M_1,M_2$ be two simply connected, connected, compact smooth manifolds without boundary and of the same dimension. Assume that $\mathfrak{X}(M_1)\cong \mathfrak{X}(M_2)$ as Lie algebras.
...
- 1,025
11
votes
1
answer
550
views
Examples of 6-manifolds without an almost complex structure
Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the ...
- 6,933