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Results tagged with 4-manifolds
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user 2356
A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different.
6
votes
Are there symplectic 4-folds with $b_+>1$, $b_-=0$?
Symplectic geography in 4 dimensions can be mapped using Chern number coordinates $(c_1^2,c_2)$. The part of the plane where $c_1^2 > 4c_2$ is uncharted. It's unknown whether there are any symplectic …
- 13.2k
17
votes
Accepted
open problems in Seiberg-Witten Theory on 4-Manifolds
One basic structural problem about the SW invariants is the question of simple type: suppose that $X$ is a simply connected 4-manifold with $b^+>1$, and $\mathfrak{s}$ a $\mathrm{Spin}^c$-structure su …
4
votes
Embeddings of $S^2$ in $\mathbb{CP}^2$
Hey Joel, long time etc. It looks to me like blowing down your knotted $S^2$ will only produce a homology 4-sphere. And one could presumably produce examples by taking some known 2-knot in $S^4$ and c …
- 13.2k
14
votes
Accepted
Slice knots and exotic $\mathbb R^4$
From Jacob Rasmussen's paper "Knot polynomials and knot homologies", arXiv:math/0504045, p.13 of ArXiv version:
Bob Gompf kindly pointed out another such application [of Rasmussen's $s$-invariant, …
- 13.2k
12
votes
Accepted
Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?
A biased answer, based on Auroux's work http://arxiv.org/abs/1003.2962.
Auroux makes a connection between bordered Floer theory and an alternative approach, due to Lekili and myself, which is (still …
- 13.2k
33
votes
What manifolds are bounded by RP^odd?
$\mathbb{RP}^3$ is the unit (co)tangent bundle to $S^2$. Thus it bounds the disc bundle in $TS^2$. Alternatively, any time you have a 2-sphere of self-intersection $\pm 2$ in a closed 4-manifold - for …
- 13.2k