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Results tagged with 4-manifolds
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user 13767
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.
17
votes
1
answer
753
views
Is there a constructive proof that in four dimensions, the PL and the smooth category are eq...
Summary
Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the …
- 5,565
13
votes
2
answers
727
views
Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?
It's easy to derive a presentation of the fundamental group of a 4-manifold if you have a Kirby diagram: The 1-handles are generators and the 2-handles are the relations. The 3- and 4-handles, which a …
- 5,565
11
votes
Accepted
Construction of invariants of 4-manifolds with the Kirby calculus
Disclaimer: Shameless self-advertising.
Yes, it can be done, and it's really beautiful! You can define the Crane-Yetter invariant with Kirby calculus, and possibly other TQFTs ("dichromatic model …
- 5,565
10
votes
1
answer
773
views
What are Kirby diagrams of candidate exotic 4-manifolds?
It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard $S^4$. The same is true for the 4-torus and several other manifolds. Handle decomposition …
- 5,565
7
votes
2
answers
793
views
How can Wall's theorem be generalised to non-simply connected manifolds?
In a sense, this is a follow-up to this question.
By work of Freedman and Wall, it is known that if two simply-connected 4-manifolds $M$ and $N$ are homeomorphic, then there is $k \in \mathbb{N}$ suc …
- 5,565
5
votes
2
answers
630
views
Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?
It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more i …
- 5,565
4
votes
2
answers
700
views
Are there Kirby diagrams with 3-handles?
Let $M\colon \partial_- M \to \partial_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M_i, i \in \{0,\d …
- 5,565
3
votes
1
answer
419
views
Simply-connected 4-manifolds can be blown up and down to complex projective planes. How abou...
There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$:
$$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# \overlin …
- 5,565
2
votes
4-dimensional TQFT with/without requiring spin structure
One of the most famous 4d TQFTs is the Crane-Yetter TQFT, or its Hamiltonian lattice formulation, the Walker-Wang model. See my question How are the Walker-Wang TQFT and the Crane-Yetter TQFT related? …
- 5,565
0
votes
Theoretical physics: Why not just $\mathbb{R}^4$?
If you accept that quantum gravity with matter should be a Topological Quantum Field Theory and that TQFTs probably can't distinguish simply connected homotopy equivalent 4-manifolds, you should come …
- 5,565