All Questions
149 questions
3
votes
1
answer
493
views
Four-dimensional vector bundles over $S^4$, intuition?
I know that $\pi_3(SO(4)) = \mathbb{Z} \oplus \mathbb{Z}$. We can choose an explicit identification as follows: given $(i, j) \in \mathbb{Z}$, we have a map $\phi: S^3 \to SO(4)$ which sends a unit ...
- 31
6
votes
1
answer
340
views
How to get a polygon from a translation surface $(X,\omega)$
Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...
- 609
0
votes
0
answers
148
views
There is no quasiregular diffeomorphism from punctured ball into ring (on the plane)
The idea is to use l2 cohomology as a quasiregular map invariant.
It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form
$f_1(x,y)dx +...
1
vote
1
answer
214
views
Chern classes of three (two) dimensional complex vector bundles
Let $M$ be a manifold.
Let $F(M,3)=\{(m_1,m_2,m_3)\mid m_1, m_2, m_3\in M, m_i\neq m_j, \text{ for any } i\neq j\}$.
Let $S_3$ be the symmetric group of order $3$.
Let $S_3$ act on $F(M,3)$ by $\...
- 1,990
1
vote
3
answers
502
views
orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates
$Z_p$:=cyclic group of order $p$.
I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates.
For ...
- 593
1
vote
0
answers
342
views
Geometric representatives of homology classes of manifolds
Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?
- 11
2
votes
2
answers
305
views
boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$
Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...
- 111
23
votes
2
answers
1k
views
fake $S^{2k}\times S^{2k}$
Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...
- 593
0
votes
1
answer
201
views
Ambient isotopy of the diagonal submanifold in product space
Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
$...
- 73
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
6
votes
2
answers
446
views
Is every $S^3$ block bundle over $S^4$ a fiber bundle?
I am interested in the difference between block bundle and fiber bundle.
Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.
A block diffeomorphism of $\Delta^p\times M$ is a ...
- 73
6
votes
0
answers
621
views
Gompf's invariant of $2$-plane fields
I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...
- 369
8
votes
1
answer
379
views
Existence of certain "nondegenerate" function and manifold topology
Let $M$ be a smooth manifold without boundary, not necessarily compact.
Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...
- 783
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
4
votes
1
answer
207
views
Singular leaf of Strebel differential
Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic ...
- 41
12
votes
1
answer
482
views
Characterize spin cobordism invariants in dimer models
The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...
- 875
1
vote
0
answers
99
views
PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
- 1,373
6
votes
1
answer
753
views
Rigidity of secondary characteristic classes
For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes
$$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$
...
- 10.4k
16
votes
1
answer
1k
views
A concrete realization of the nontrivial 2-sphere bundle over the 5-sphere?
Since $\pi_4 (PU(2)) = \pi_4 (SO(3)) = {\mathbb Z}_2$, the two-element group,
we know that half of the two-sphere bundles over the 5-sphere $S^5$ are trivial
and the other half are non-trivial and ...
- 8,336
12
votes
3
answers
1k
views
A version of Lusternik–Schnirelmann category for good open covers
Recall that the Lusternik–Schnirelmann category (or LS-category) of a space is the integer $n$ such that there is an open cover by $n+1$ open sets which have nullhomotopic inclusions, and no such ...
- 35.5k
2
votes
1
answer
214
views
winding number for outer-pointing normal
While trying to characterize the complexity of a closed differentiable curve (for a path planning application), I've been using a notion which is similar in spirit to the winding number of a curve. ...
- 121
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
28
votes
1
answer
2k
views
Example of 4-manifold with $\pi_1=\mathbb Q$
This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.
- 2,623
9
votes
1
answer
1k
views
Classification of higher dimensional manifolds
It is known that a 2-connected closed smooth 6-manifold is homeomorphic to S^{6}
or connected sum of (S^{3}xS^{3}). My question is whether we have a similar statement for (n-1)-connected closed ...
- 399
4
votes
1
answer
326
views
Reference question: space of embeddings
Let $X$ be a compact surface of genus $g \geq 1$. Then it is a well known fact that the space of embeddings into $\mathbb{R}^{\infty}$ is contractible. The proof uses Whitney's embedding theorem. ...
- 471
16
votes
3
answers
3k
views
open problems in Seiberg-Witten Theory on 4-Manifolds
What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or ...
22
votes
2
answers
1k
views
Eversion of the 6-sphere in 7-space
Say that $S^n$ "admits eversion" if the inclusion $S^n \rightarrow \mathbb{R}^{n+1}$ is regularly homotopic to the antipodal map (where a "regular" homotopy is a continuous path through immersions).
...
- 732
1
vote
2
answers
897
views
Can a sphere be a phase space?
Put in other words, given an even-dimensional sphere $S^{2k}$: is there a manifold $M$ such that $T^* M$ is diffeomorphic to $S^{2k}$?
- 13
13
votes
3
answers
1k
views
Manifold whose universal covering is a sphere but which is not a space form?
Let $M^n$ be a smooth manifold whose universal cover is homeomorphic $\mathbb{S}^n$, are there examples where $M^n$ is not homeomorphic to a space form ?
The answer may vary if you replace ...
- 4,123
9
votes
2
answers
367
views
Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?
Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?
It was proved in "On the geometric ...
- 2,623
9
votes
1
answer
5k
views
Manifolds are paracompact
By Definition, smooth manifolds are assumed to be Hausdorff and to satisfy the second countability axiom.
I have heard (but never seen written) that these assumptions imply paracompactness (and thus ...
- 10.4k
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
8
votes
3
answers
3k
views
Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?
Hello,
I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the ...
- 83
1
vote
1
answer
242
views
The inertia subgroup of `$\Theta_n$` for Lie groups
See Smooth manifold with non-trivial inertia group? (wrt homotopy spheres) for the definition of $\Theta_n$ and inertia subgroups.
I'm wondering what can be said about Lie groups. If $M^n$ is an n-...
- 732
7
votes
1
answer
768
views
More on completion/compactification of open manifolds
This is a follow up to one of my previous questions (81714). Suppose that $M$ is an open manifold, say with a single end. Previously, I was concerned with realizing $M$ as the interior of a compact ...
- 21.5k
11
votes
1
answer
620
views
Is $SL(n,\mathbb{Z})$ a CAT(0) group?
Is it possible to find a CAT(0) space on which the matrix group $SL(n,\mathbb{Z})$ acts properly discontinuously and cocompactly? Note: when the cocompactness is dropped , it is possible.
- 1,373
23
votes
2
answers
2k
views
Uniqueness of compactification of an end of a manifold
Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-...
- 21.5k
11
votes
2
answers
811
views
Higher dimensional Heegaard splittings?
Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing ...
- 732
5
votes
0
answers
717
views
Can Euler Class be defined by the Splitting Principle for Real Vector Bundles?
Let $M$ be a manifold and $S$ its sphere bundle with fiber $\mathbb{S}^n$. As we know, the notion of the Euler class is raised from the problem of finding a global form on $S$ which restricts on each ...
- 643
4
votes
1
answer
1k
views
Perfect Morse functions
Does anyone know a concrete example of a Morse function on some manifold that is perfect with respect to some field but not with respect to $\mathbb Z_2$?
2
votes
1
answer
3k
views
Does $S^2$ have a trivial normal bundle in any closed orientable manifold?
We know that the middle circle $S^1$ in Mobius band has a nontrivial normal bundle. Now consider the higher dimensional case. Let $M$ be a $n$-dimensional ($n\geq5$) closed orientable manifold and ...
- 1,373
17
votes
3
answers
1k
views
Codimension zero immersions
Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?
Remark: If the sphere had dimension k smaller than n-1, then such an immersion ...
- 10.4k
8
votes
2
answers
2k
views
Why is the mapping class group of hyperbolic manifolds finite?
Hi! I'm trying to understand why a hyperbolic n-manifold has finite mapping class group if $n \geq 3 $. In books I'm reading it's said it's a consequence of Mostow's rigidity theorem:
"If M and N are ...
- 425
3
votes
1
answer
1k
views
Orientation of a "glued"-manifold
Im wondering if there's a short way to prove that when two manifolds with diffeomorphic boundaries are glued together along the boundaries the orientations of these must be inverse to each other. That ...
- 947
3
votes
1
answer
552
views
Is the Action of the mapping class group transitive on embedded arcs?
Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The ...
- 27.5k
7
votes
1
answer
456
views
Space-discriminating injective curve
Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
- 5,053
3
votes
0
answers
446
views
When does the normal bundle of a submanifold of Euclidean space admit a flat connection?
Given a smooth submanifold of $R^n$, I was wondering if there is a reasonably simple criterion for deciding whether its normal bundle admits a flat connection. I am not ruling out monodromy in the ...
- 313
23
votes
4
answers
5k
views
De Rham decomposition theorem, generalisations and good references
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
- 28.9k
15
votes
2
answers
973
views
Infinity de Rham quasi-isomorphism
This question is similar to Do chains and cochains know the same thing about the manifold? in the sence that both deal with a natural "comparison" quasi-isomorphism that does not preserve the ring ...
- 23.5k