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Results tagged with differential-topology
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user 1573
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”.
9
votes
Locally conformally flat
Any open parallelizable $n$-manifold immerses into $\mathbb R^n$ (this is special case of by Hirsch-Smale h-principle), and hence is conformally flat.
There are many examples of conformally flat man …
- 29.1k
32
votes
Accepted
Is the minimal volume a topological invariant?
Minimal volume is not a homeomorphism invariant. It is shown in [L. Bessières, Un théorème de rigidité différentielle, Comm. Math. Helv. 73 443-479 (1998)] that the minimal volume of the connected sum …
- 31.5k
9
votes
Computation on characteristic classes
The papers "Characteristic Classes and Homogeneous Spaces, I, II" by A. Borel and F. Hirzebruch is a classical resource. In general, homogeneous spaces form a rich class of examples to compute and pla …
- 29.1k
7
votes
Accepted
Transitive action on non-orientable $ M $ lifts to orientable double cover
There is a general theory for lifting Lie group actions to covering spaces, see Bredon's monograph "Introduction to compact transformation groups", chapter 1, section 9. In the case of orientation cov …
- 29.1k
4
votes
Counting connected manifolds
There are countably many compact topological manifolds, as was shown by Cheeger-Kister in Counting topological manifolds, Topology 9 (1970) 149–151, https://doi.org/10.1016/0040-9383(70)90036-4. The p …
- 35.5k
3
votes
Accepted
Hyperbolization with word-hyperbolic fundamental group
Charney-Davis in Strict hyperbolization showed how to make $N$ locally CAT($-1$), provided $M$ is PL.
Ontaneda in Riemannian hyperbolization showed how to make $N$ a Riemannian manifold of negative se …
- 29.1k
28
votes
Accepted
Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such...
The answer is no. If $L$, $L^\prime$ are 3-dimensional lens spaces and $S^1\times L$ is diffeomorphic to $S^1\times L^\prime$, then the covering space of $S^1\times L$ corresponding to the torsion sub …
- 29.1k
13
votes
Accepted
When is a bi-Lipschitz homeomorphism smoothable?
Any self-homeomorphism of a manifold of dimension $\neq 4$ is topologically isotopic to a bi-Lipschitz homeomorphism, see lemma 2.4 in Lipschitz and quasiconformal approximation of homeomorphism pairs …
- 29.1k
11
votes
Accepted
Obstruction to a general S^1-action
V. Puppe, in Simply connected manifolds without $S^1$-symmetry. Algebraic topology and transformation groups, Proc. Conf., Göttingen/FRG 1987, Lect. Notes Math. 1361, 261-268 (1988) proved the followi …
- 4,195
4
votes
Accepted
Homogeneous manifold deformation retracts onto compact submanifold
Mostow-Karpelevich theorem says that if $G/G^\prime$ is a homogeneous space where $G$, $G^\prime$ are Lie groups with finitely many connected components, and maximal compact subgroups $K\supset K^\pri …
- 29.1k
11
votes
Accepted
Piecewise linear Poincaré conjecture
For spheres of dimension $n>5$ the PL Poincare conjecture follows from the s-cobordism theorem. Indeed, removing disjoint two small open disks one gets an s-cobordism (this uses excision in homology, …
- 29.1k
11
votes
Quotient of arbitrary free involution on $S^n$
In every dimension $\ge 4$ there is a fake real projective space, i.e., a manifold that is homotopy equivalent but not homeomorphic to $RP^n$.
Here you can find a computation for the topological surg …
- 29.1k
6
votes
Accepted
Diffeomorphism type of the added sphere in simply connected surgery
The ambiguity in the the added homotopy sphere is captured by a suitable inertia group $I(X)$, which in this case is the group of homotopy spheres $\Sigma$ such that $\Sigma$ bounds a parallelizable m …
- 29.1k
33
votes
Accepted
Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^...
A closed smooth $n$-manifold embeds into $\mathbb R^{2n-1}$ if and only if the normal $(n-1)$th Stiefel-Whitney class vanishes. This is due to Hirsch-Haefliger in dimensions $\neq 4$ and to Fang in di …
- 28k
13
votes
Why is the first integral Pontryagin class a homeomorphism invariant?
For spin manifolds this is proved in Corollary 1.22 (p.17) of Kammeyer's Diploma. Kreck's claim that the "spin" assumption can be dropped is mentioned after the corollary, with the caveat that "the au …
- 29.1k