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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”.
51
votes
Accepted
Can a topological manifold have different tangent bundles?
This is answered in [Crowley, Diarmuid J.; Zvengrowski, Peter D, On the non-invariance of span and immersion co-dimension for manifolds, Arch. Math. (Brno) 44 (2008), no. 5, 353–365], see here.
Spec …
- 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 …
- 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 …
- 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
26
votes
Accepted
Are there non-smoothable homotopy/homology spheres?
Since the Poincaré conjecture is known in all dimensions any homotopy $n$-sphere is homeomorphic to $S^n$ and hence admits a smooth structure.
Any manifold of dimension $\le 3$ admits a smooth struct …
- 29.1k
18
votes
Accepted
Does every smooth manifold of infinite topological type admit a complete Riemannian metric?
By Whitney embedding theorem any smooth manifold embeds into some Euclidean space as a closed subset. The induced metric is complete.
In fact, a good exercise is to show that any Riemannian metric is …
- 29.1k
16
votes
Accepted
representatives of the group of homotopy 7-spheres
This is done in the paper "An invariant for certain smooth manifolds" by James Eells and Nicolaas Kuiper. They introduce and study the so called $\mu$-invariant which is strong enough to classify homo …
- 29.1k
13
votes
Accepted
Topology of function spaces?
Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent Oberwolfach report from a meeting on the subject.
If you share what $X,Y$ you are interested in, I may be …
- 29.1k
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
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
12
votes
Accepted
Smooth structures on the connected sum of a manifold with an Exotic sphere
Surgery theory provides a framework for classifying closed higher-dimensional manifolds, but unfortunately, a definitive classification is known only for a very few homotopy types. Here is how surgery …
- 29.1k
12
votes
When does the tangent bundle of a manifold admit a flat connection?
If a vector bundle admits a flat connection, then the rational Pontryagin classes of the tangent bundle vanish (as follows from Chern-Weil theory, see Milnor-Stasheff's "Characteristic classes", Appen …
- 29.1k
12
votes
Accepted
Homotopy equivalence of diffeomorphism groups
I am not aware of a detailed reference but here is a sketch.
$\mathrm{Diff}^r(M)$ is a Hilbert manifold (i.e., it is locally homeomorphic to a separable Hilbert space). This can be found e.g., in s …
- 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
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 …
- 29.1k