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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).
5
votes
Accepted
Embeddings of magnetic cotangent bundles over surfaces into closed symplectic 4-manifolds
This can always be done.
Let's first treat the case when $\Sigma$ is not a torus. Then take any symplectic $4$-manifold $(M,\omega)$ where $\Sigma$ can be embedded as a Lagrangian surface. Now, take a …
- 28.9k
9
votes
Accepted
Arbitrary torsion in cohomology of Kähler manifolds
The answer is positive and can be deduced from Proposition 15 of "Sur la topologie des varietes algebriques en characteristique p" by Serre. According to this proposition for any finite group $G$ ther …
- 28.9k
6
votes
Homotoping diffeomorphism to a $J$-holomorphic one
No. Take the torus $T^2=\mathbb R^2/\mathbb Z^2$ and consider the self-map induced by the matrix
$$\begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}$$
PS. As for simply-connected examples, I think that a …
- 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
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
0
votes
Accepted
Non-algebraic Kähler threefolds with abelian $\pi_1$ of arbitrarily large rank
Let's construct such a Kahler $3$-fold $X$. It will be obtained as an elliptic fibration over a projective surface $S$ with abelian fundamental group $\mathbb Z^{2g}$.
Construction. Recall first that …
- 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
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
8
votes
Accepted
Fano manifold becoming general type upon conjugation
No. $(M,J)$ and $(M,-J)$ have conjugate pluri-canonical rings, hence have same Kodaira dimension.
Proof. Take a section $\mu$ of $K_{(M, J)}^{\otimes n}$, then $\bar \mu$ is a holomorphic section of $ …
- 28.9k
8
votes
Accepted
Complex projective manifold with an antiholomorphic involution
Corrected. As Robert Bryant points out, it is not enough to show that the manifold can be realised as a submanifold of $\mathbb CP^n$ invariant under some anti-holomorphic involution of $\mathbb CP^n$ …
- 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
2
votes
Accepted
Topological factors of complex projective manifolds
Let $M=\mathbb CP^2\#\mathbb CP^2$ and let $S=T^2$ be the $2$-dimensional torus. I think this gives an example for the original question. As for the symplectic version of the question, I am sure it is …
- 28.9k
6
votes
Accepted
Diffeomorphisms pushing forward vector field to any non-zero multiple
Such a manifold exists. First let's construct a non-compact example.
Take $PSL(2,\mathbb R)$ and take two $1$-parameter subgroups, given by $$\begin{pmatrix}
e^{t} & 0 \\
0 & e^{-t}
\end{pmatrix}, \ …
- 39.1k
8
votes
Accepted
Real points on a projective curve
It looks like the lower bound is $0$ if $d$ is even and $1$ if $d$ is odd.
Construction. Suppose $d=2p$. Take the curve $F=(z_1^2+z_2^2+z_3^2)^p=0$. It doesn't have real points at all. Taking a small …
- 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 …
- 4,745