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Results tagged with differential-topology
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user 66688
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
11
votes
2
answers
311
views
Cohomology of foliations and closed forms along the leaves
Let $M$ be a manifold equipped with a codimension one, transversely orientable, regular foliation $F \subset M$. Let $\alpha\in \Omega^k(M)$ be a differential form on $M$ that is not closed on $M$ b …
- 2,818
6
votes
2
answers
814
views
Lifting sections of a projective bundle to a vector bundle
Let $E\to M$ be a smooth $\mathbb{K} = \mathbb{R}, \mathbb{C}$ - vector bundle over a possibly non-compact connected manifold $M$. Denote by $\mathbb{P}(E) \to M$ its projectivization, which is obtain …
- 2,818
9
votes
0
answers
341
views
Diffeomorphism type of Ricci-flat four manifolds
Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows:
A) Is there a classification of the possible homeomorphism types of …
- 2,818
8
votes
1
answer
847
views
Tangent space of the space of smooth sections of a bundle
Let $E\to M$ be a real vector bundle of finite rank over a closed differentiable manifold $M$. Let $C^{\infty}(E)$ denote the space of smooth sections of $E$ and let $e\in C^{\infty}(E)$ be a section. …
- 2,818
5
votes
2
answers
1k
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Triviality of the adjoint and endomorphism bundles
Let $P$ be be a principal bundle over a manifold $M$ with structure group $G$, where $G$ is a Lie group. Let $E = P\times_{\rho} \mathbb{R}^{k}$ be a vector bundle associated to $P$ through a faithful …
- 2,818
10
votes
2
answers
719
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 $[\a …
- 2,818
10
votes
1
answer
844
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On the topology induced by a Lorentzian metric
Let $(M,g)$ be a time-oriented smooth Lorentzian manifold, with Lorentzian metric $g$. In the following thread:
Lorentzian distance induced topology(a.k.a. Interval topology)
physicist @ValterMoretti …
- 2,818
4
votes
1
answer
358
views
Compatible reductions of the structure group of a principal fiber bundle
Let $P$ be a principal bundle over a manifold $M$ with structure group the Lie group $G$. Assume that $P$ admits to distinct topological reductions, say $Q_{1}$ and $Q_{2}$, where $Q_{a}$, $a=1,2$ are …
- 2,818
2
votes
1
answer
619
views
Homotopy type of an oriented, closed, simply connected manifold
It is well known that every closed, oriented, simply-connected four-manifold $M$ is homotopy equivalent to a CW-complex consisting on a 0-cell, a wedge of two spheres and a 4-cell.
I was wondering i …
- 2,818
16
votes
2
answers
1k
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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 thei …
- 2,818
5
votes
1
answer
2k
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The space of homotopy classes of maps of products of spheres
Proposition 17.6.1 of "Differential form in Algebraic Topology" by Bott and Tu proves the following beautiful result:
$[S^{q}, X]\simeq \frac{\pi_{q}(X,x)}{\pi_{1}(X,x)}$
where $S^{q}$ is the $q$-sp …
- 2,818
7
votes
1
answer
346
views
Differential geometry without the Hausdorff condition or the second axiom of countability
I would like to know how the standard differential geometry of manifolds would change if we didn't assume the Hausdorff condition and/or the second axiom of countability. There are some simple things …
- 2,818