All Questions
30 questions
5
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122
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Explicit parallelization of an exotic sphere
I asked this question on MathStackExchange a week ago (see here), but, despite a few upvotes, I received no comments or answers. Ideally, I would love a detailed answer, but a yes/no would do the job! ...
- 51
1
vote
0
answers
240
views
Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
- 356
0
votes
0
answers
232
views
What is the adjoint bundle of groups $P\times_{G}G$?
It is said that G acts on itself by conjugation. I am familiar with another type of adjoint bundle in which a representation of G on a vector space is given. Can someone explain the differences and ...
- 35
0
votes
0
answers
329
views
Pushforwards in vector bundles over a topological spaces
I have been reading the discussion from Pushforward and pullback..
I understand that it is quite straight forward to construct a pullback of a vector bundle. In the discussion it is clear that if we ...
- 615
5
votes
2
answers
361
views
Exterior differentiation of foliations
Let $M$ be a differentiable manifold.
Let $T^*M$ be the cotangent bundle of $M$.
Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\...
- 123
5
votes
0
answers
121
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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
7
votes
0
answers
270
views
The Todd class and Weyl's character formula
Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...
- 1,379
1
vote
0
answers
246
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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
4
votes
1
answer
305
views
Classification of functorial smooth vector fiber bundles
Let $\mathrm{Bundle}$ be the category whose objects are smooth vector fiber bundles over $\mathbb{R}$, and morphisms are fiberwise smooth linear map (that is, the base is not assumed to be fixed).
Let ...
- 6,962
2
votes
0
answers
255
views
Extending an embedding with trivial normal bundle
I am recently studying the book Notes on Cobordism Theory by R. E. Stong and I have noticed that the proposition below is (implicitly) used (for example to extend a $(B,f)$ structure on a boundary of ...
- 131
4
votes
1
answer
334
views
Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
It seems that there is no digital copy of Leon Karp's Ph.D. thesis
L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief ...
- 4,195
1
vote
1
answer
729
views
Vector field tangent to a submanifold and transverse to the zero section
In Hirsch's Differential Topology there's the following :
Suppose a compact $n$-manifold can be expressed as $A\cup B$ where $A,B$ are compact $n$-dimensional submanifolds and $A\cap B$ is an $(n-1)$-...
- 791
2
votes
1
answer
182
views
Vector field along an immersion whose covariant derivative is the differential
Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one ...
- 2,247
5
votes
0
answers
311
views
Hopf fibration extended to bundle over $\mathbb{C}^2$
Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...
5
votes
1
answer
413
views
Shrinking and stretching of vector bundles
Let $M$ be a manifold, $p:E\to M$ a rank $d$ vector bundle. Suppose that $U \subset E$ is an open subset such that $U \cap p^{-1}(x)$ is nonempty and convex for all $x \in M$. Is it true that $U \to M$...
- 40.2k
1
vote
0
answers
61
views
Minimal radius of a ball admitting a trivialization of a vector bundle
Let $X$ be a compact Hausdorff space and $p : V \to X$ a complex vector bundle of rank $n$. For $r > 0$ let $B(r,x)$ denote the open ball of radius $r$ around $x$. Does there exist an $r$ such that,...
5
votes
1
answer
379
views
Conversion formula between "generalized" Stiefel-Whitney class of real vector bundles: O(n) and SO(n)
$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$,
$$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$
Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as:
$$
w_j(...
- 10.5k
6
votes
0
answers
291
views
Can we "Curve" a manifold, as much as possible?
Assume that $M$ is a $k$ dimensional manifold which is embedded in $\mathbb{R}^n$. We define the map $\phi_{M,n}: M \to G(k,n)$ with $\phi_{M,n} (x)= T_x M$, the tangent space to $M$ at point $x\in M$....
- 356
2
votes
0
answers
436
views
Why is $\Omega_k(C^\infty(M))\to\Omega^1(M)$ surjective?
Let $M$ be a smooth manifold and let $A=C^\infty(M).$
We consider module of Kahler differentials $\Omega_k(A)$ and module of 1-forms $\Omega^1(M).$ Denote Kahler differential by $d_k$ and classical ...
- 1,615
0
votes
1
answer
195
views
Triviality of certain vector bundles
Let $M$ be a smooth manifold and let $SM$ be the bundle of symmetric bi-linear forms on $TM.$ Riemannian metrics are a particular kind of sections in this bundle. Since any manifold admits a global ...
- 463
5
votes
2
answers
1k
views
Vector bundles, finitely generated projective module?
Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...
- 151
2
votes
1
answer
827
views
Tangent bundle of $S^2 \times S^1$ trivial or not [closed]
Is the tangent bundle of $S^2 \times S^1$ trivial or not?
user78817
1
vote
1
answer
396
views
Orientability of Surfaces and the Fundamental Group [closed]
Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...
- 11
1
vote
0
answers
414
views
How to show the Euler Characteristic is equal to self-intersection number of zero-section [duplicate]
myThe definition of the Euler characteristic (given in Guillemin and Pollack's "Differential Topology") of a compact oriented manifold $X$ is the self-intersection number of the diagonal $\Delta$ in $...
- 111
1
vote
1
answer
384
views
A version of implicit function theorem when sections are not everywhere smooth?
Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$
a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section
...
- 3,245
2
votes
0
answers
966
views
Can one always extend a smooth section defined on a non compact submanifold to the whole manifold, provided it extends continuously to the closure?
Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$
(without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed.
Suppose $s: X \...
- 3,245
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 ...
- 1,236
84
votes
4
answers
6k
views
Parallelizability of the Milnor's exotic spheres in dimension 7
Are the Milnor's seven dimensional exotic spheres parallelizable?
- 1,236
3
votes
3
answers
990
views
Topology of maps between fibers of vector bundles
I'm in doubt about the topology of maps between fibres of vector bundles.
Consider $E$ and $F$ vector bundles and the set of all linear maps from a fibre of $E$ to a fibre of $F$, ie, the set of all ...
- 121
34
votes
10
answers
11k
views
Why is cotangent more canonical than tangent?
You don't need a metric to define the differential of a function,
and the cotangent bundle carries a canonical one-form.
But you do need a metric to define the gradient, and the
tangent bundle does ...
- 3,267