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Results tagged with gt.geometric-topology
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user 943
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
23
votes
Gromov's list of 7 constructions in differential topology
Unfortunately I missed the talk, but on the other hand Gromov have just produced a new paper called
Manifolds : Where do we come from ? What are we ? Where are we going ?
It can be found on his we …
- 28.9k
22
votes
Accepted
Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$?
This is a notorious open problem. For the moment the simplest compact four-manifold that is announced to admit (infinite number of) exotic smooth structures is $S^2\times S^2$. This result is containe …
- 28.9k
22
votes
Accepted
When does the tangent bundle of a manifold admit a flat connection?
The question of existence of flat connection on tangent bundles of manifolds was studied quite extensively. Milnor proved in one of his early papers that surfaces (compact without boundary) of non-zer …
- 28.9k
19
votes
Accepted
Unique almost complex structure up to diffeomorphism
You were lucky to find the only possible example. If you take any manifold of dimension $\ge 4$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrab …
- 28.9k
17
votes
Accepted
Is $Sym^g$ of a Riemann Surface of genus $g$ Calabi-Yau?
This is not a Calabi-Yau if $g\ne 1$ (for any definition of Calabi-Yau). Indeed, there is a degree one map from the symmetric power of the curve to a torus of dimension $g$. Pull-back of the volume f …
- 28.9k
16
votes
Closed 3-manifolds with free abelian fundamental groups
Only $\mathbb Z$ and $\mathbb Z^3$ (for $T^3$) are free abelian groups that appear as fundamental groups of $3$-manifolds. Hopefully the following is an approximative proof.
The manifold must be prim …
- 28.9k
13
votes
Study topology from existence of multiple smooth structures?
Since this question might have many answers, I can propose one possible answer (I hope this is really an answer to the question.) So, suppose we have a smooth $4$-dimensional manifold. We want to tria …
- 28.9k
13
votes
Classification problem for non-compact manifolds
This answer complement the answers of Henry and Algori. I think, it is worth to strees, that a classification of open manifolds does not follow from a classification of compact manifolds. Open surface …
- 28.9k
11
votes
Accepted
Does a triangulation without fixed simplex property always exist?
EDITED. The arugment related to Mostov rigidity is completed according to a nice suggestion of Tom Church
The answer to the first question is no. There exsit manifolds of dimension 3 such that every …
11
votes
Accepted
Chern/Hodge numbers of the conjugate complex manifold
These are all the same.
As for Hodge numbers, you can choose a Kahler metric $g$ on $(M,J)$, and it will also be Kahler for $(M,-J)$. Now we know that $h^{p,q}$ is the dimension of the space of harmon …
- 28.9k
11
votes
Accepted
Osculating conics and cubics and beyond
These highly osculating curves were studied, in particular by V.I. Arnol'd. One of the important references will be:
Topological invariants of plane curves and caustics.
Dean Jacqueline B. Lewis Me …
- 28.9k
10
votes
Accepted
Antiholomorphic involution with a fixed point
No. There exist both non-algebraic and projective counterexamples.
1 Non-algebraic example. Take a flat Euclidean torus $T^4=M$ and let $Z$ be its twistor space. It has an antiholomorphic involution w …
- 28.9k
10
votes
Accepted
Deformation equivalent vs diffeomorphic to projective manifold
I believe the answer is yes and follows from the combination of Theorem 4.6 here https://arxiv.org/pdf/math/0111245.pdf
and Theorem 1.3 here https://arxiv.org/pdf/math/0111245.pdf
The first result sho …
- 28.9k
10
votes
Symplectic structure on the square of a 3-manifold
Here is a Kahler example. Consider a hyper-elliptic curve $C$ of positive genus with involution $\sigma$. Take $C\times S^1$ and quotient $C\times S^1$ by $\mathbb Z_2$ that rotates $S^1$ by a half-tu …
- 28.9k
9
votes
Classification of symplectic surfaces
This is not a complete answer to the question, and I don't know if a complete answer is written down anywhere in the literature. In the first revison of the answer I tried to adress all the 10 commen …
- 28.9k