Questions tagged [4-manifolds]
Questions specifically about $4$-dimensional manifolds
85
questions
1
vote
0
answers
29
views
Kirby Diagram of Enriques Surface
I would like a Kirby/handle diagram of the Enriques surface, and I whilst I haven't been able to find one in the literature, there is this diagram (originally due to Kondo I believe, though taken here ...
- 11
2
votes
1
answer
63
views
Surgery on smooth four-manifold preserves the intersection form
In these notes (p. 190), it is claimed that (possibly with superfluous hypotheses):
Claim. Let $M$ be a closed connected oriented smooth 4-manifold and let $c\colon S^1\hookrightarrow M$ be an ...
- 356
0
votes
0
answers
18
views
Lefschetz fibration on product of surfaces
If I understand the literature correctly, then every symplectic 4 manifold (potentially up to connected sum with complex projective space) admits a lefschetz fibration with codomain a two sphere.
In ...
- 1,455
0
votes
0
answers
50
views
Orthogonal vectors in 4D
Consider two 4D vectors: $v_1=(\cos\varphi_1\sin\theta_1\sin\psi_1,\sin\varphi_1\sin\theta_1\sin\psi_1,\cos\theta_1\sin\psi_1,\cos\psi_1)$ and $v_2$, this vectors are orthogonal $v_1 \cdot v_2=0$, I ...
0
votes
1
answer
109
views
$\mathbb{C}P^2$ is not diffeomorphic to $\overline{\mathbb{C}P^2}$
I am working through 4-Manifolds and Kirby Calculus by Stipsicz and Gompf. At the beginning of Section 1.3, they have a list of exercises regarding $\mathbb{C}P^n$ and $\mathbb{R}P^n$. The part I ...
- 524
1
vote
0
answers
35
views
Knot Trace and Trace Lemma 2
I recently asked a question about the proof of theorem 1.8 in Miller and Picorillos paper:
Knot Traces and Trace Lemma
I will continue to keep the same notation as in my previous question.
Following ...
- 269
5
votes
1
answer
105
views
Knot Traces and Trace Lemma
I'm currently reading the paper by Miller and Picorillo titled Knot Traces and Concordance. I have some confusion about the proof of theorem 1.8. In particular the 'if' direction.
I'll restate some of ...
- 269
5
votes
1
answer
263
views
Every knot is slice, find the error in the following argument
It is well known that exist knots which are not slice, however I came up with the following argument contradicting this fact and I fail to see where the mistake is, can somebody help me?
Let $K_0$ ...
- 5,498
1
vote
1
answer
58
views
Picturing twisting of strands explicitly after blow downs
In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...
- 183
2
votes
0
answers
118
views
Non-orientable surfaces embedded in 4-manifolds
Question: Are there not-so-trivial statements that hold for non-orientable surfaces embedded in 4-manifolds, but are known not to hold (or do not make sense) for orientable ones?
Some context. I am ...
- 1,226
7
votes
0
answers
144
views
Showing that a 4-manifold obtained by attaching a 2-handle is simply-connected
I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation:
Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...
- 2,076
0
votes
2
answers
95
views
Action of the Gauge Group on the Configuration Space of Seiberg-Witten Theory
In Seiberg-Witten theory, the group of gauge transformations is $Map(M,S^1)$.
For a configuration $(A,\psi)$, where $A$ is a unitary connection on the determinant line bundle, $\psi$ is a spinor, and $...
1
vote
1
answer
49
views
Does the 3-manifold $S^1\times S^2$ bound a smooth integral homology ball?
Does the 3-manifold $S^1\times S^2$ bound a smooth (integral) homology ball?
The only 4-manifolds I know whose boundary is $S^1\times S^2$ are $S^1\times D^3$ and $D^2\times S^2$, and both are not ...
- 2,076
2
votes
1
answer
113
views
Topological classification of complex surfaces
The famous Enriques–Kodaira classification classfies the minimal complex surfaces by algebraic invariants, which give tight restrictions on the topology of the underlying manifolds. What about the ...
- 1,895
1
vote
0
answers
44
views
Is Homeo(M) locally path-connected for a general topological manifold?
I am wondering if there exists a closed topological manifold for which Homeo$(M)$ is not locally path-connected. If $M$ admits a smooth structure, then one can prove that Homeo$(M)$ is in fact locally ...
- 473
1
vote
0
answers
46
views
Categories (other than TOP) to rule out exotic $\mathbb{R}^4$?
I'm a non-mathematician who is interested in differential topology.
If I understand correctly, the existence of exotic $\mathbb{R}^4$ is directly linked to the failure of smooth h-cobordism theorem ...
- 755
7
votes
3
answers
172
views
If $n>1$, the square of the odd Fibonacci number $F(2n+1)$ can be written as the sum of exactly $F(2n+1)+1$ nonzero squares.
While reading a paper by Owens (arXiv:1906.05913) about embeddings of rational homology balls in the complex projective plane, I found out the following somewhat unexpected number theory corollary (...
- 490
4
votes
1
answer
109
views
Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form?
Consider the smooth four-manifold $M = (S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$. Does $M$ admit a symplectic form?
If $\omega$ is a symplectic form, then the real cohomology class $[\omega]$...
- 96.9k
2
votes
1
answer
201
views
Intersection pairing of weighted projective plane
Let $a,b,c$ be mutually relatively prime positive integers. The weighted projective plane $X:=\Bbb CP^2(a,b,c)$ is the quotient space $\Bbb C^3-\{0\}/(z_1,z_2,z_3)\sim (\lambda^a z_1,\lambda^b z_2,\...
- 2,076
2
votes
0
answers
133
views
Is there a relation between self-intersection and covering map?
Let $X$ and $Y$ be compact oriented smooth 4-manifolds. There is a well-defined intersection form $H_2(X)\times H_2(X)\to \Bbb Z$, $(\alpha,\beta)\mapsto \alpha \cdot \beta$, and similarly for $Y$ (...
- 2,076
0
votes
0
answers
46
views
Need to calculate the 3D volume of a growing 4D hypersphere in Minkowski space
4D geometry and Minkowski space are areas of expertise which I fundamentally lack, so I'm hoping people are able to help me with this.
The problem is this, if you had a hypersphere in Minkowski space ...
- 53
1
vote
1
answer
171
views
Intersection form of a 4-manifold with boundary
For a closed oriented 4-manifold $X$, the bilinear intersection form $H_2(X)\times H_2(X)\to \Bbb Z$, $(a,b)\mapsto \langle PD(a)\cup PD(b), [X]\rangle$ is unimodular, which can be shown by Poincare ...
- 2,076
3
votes
1
answer
130
views
Some questions about Twistor Space of a closed $4$-manifold
Let $(M,g)$ be a closed Riemannian manifold of dimension $4$. We denote its twistor space, the space of almost complex structures on the tangent bundle $TM$ by $Z\xrightarrow{\pi}M$. At any point the ...
- 1,307
2
votes
0
answers
59
views
Circle action on a 5-dimensional manifold and Euler class
A pseudofree $S^1$-action is a smooth $S^1$-action on a smooth $(2n+1)$-manifold such that the action is free except for finitely many exceptional orbits with isotropy $\Bbb Z_{a_1},\dots,\Bbb Z_{a_k}$...
- 2,076
2
votes
0
answers
40
views
$c_1^2(L)\geq 0$ for a nef line bundle $L\to S$ over a complex surface
Let $S$ be a compact complex surface and $L\to S$ a holomorphic line bundle. $L$ is said to be nef if $c_1(L)[C]\geq 0$ for any curve $C\subset S$. Is it true that, for nef $L$, we have $c_1^2(L)\geq0$...
- 2,076
0
votes
0
answers
94
views
Kirby Diagram for the Trefoil
I'm studying R.E. Gompf and A.I. Stipsticz, 4-Manifolds and Kirby Calculus and I got stuck with a question. Let $K$ be the right-handed trefoil embedded in $\partial \mathbb{D}^4$, we know that, ...
- 105
0
votes
0
answers
57
views
How to visualize these shapes $\#^k(S^2 \times S^2)$?
How can I visualize (i.e. "draw") these shapes $\#^k(S^2 \times S^2)$ the $k$-fold connected sum of the product of two spheres.
This was called the connected sum of two projective planes. ...
- 24.2k
9
votes
1
answer
340
views
Relatively minimal elliptic surfaces which are not minimal
A complex surface is called minimal if it contains no $(-1)$-curves, while an elliptic surface $\pi : X \to C$ is called relatively minimal if the fibers of $\pi$ contain no $(-1)$-curves. It follows ...
- 96.9k
3
votes
0
answers
70
views
Expressing complexified tangent bundle of a spin 4-manifold as a Hom bundle
I am reading Moore's book Lectures on Seiberg-Witten Invariants, section 2.2. First here are some defintions that the book uses.
The group $\operatorname{Spin}(4)$ is defined to be the product group $...
- 2,076
4
votes
2
answers
171
views
Killing a cohomology class in a manifold by taking out a submanifold representing its Poincaré-dual
Let $M$ be an oriented $n$-manifold, $\alpha \in H^k(M;R)$ a cohomology class so that its Poincaré-dual $a\in H_{n-k}(M;R)$ is represented by an embedded $(n-k)$-manifold $F$.
Question: Is the ...
- 378
0
votes
0
answers
128
views
Homology of orientable 4-manifold given fundamental group and Euler characteristic
I'm working through old topology quals and came across this question:
Let M be a closed, connected, orientable 4-manifold with fundamental group $π_1(M) = \mathbb{Z}_3 ∗ \mathbb{Z}_3$ and Euler ...
- 1
4
votes
1
answer
305
views
Notes on Low-Dimensional Topology
I am studying algebraic topology at the moment and I'm halfway done with Hatcher's book. I am extremely interested in low-dimensional topology, so I was wondering if anybody knows a good set of notes ...
- 93
1
vote
1
answer
61
views
Conflicting definitions of instanton number
In Nakahara's "Geometry, Topology and Physics" (and many other sources) the instanton number of an $SU(2)$ instanton $A$ with curvature $F^A$ is defined by
$$\int_{S^4}\text{ch}_2(E)=\frac{1}...
- 2,466
3
votes
1
answer
78
views
Representation by Complex Surfaces
Consider the topological manifold $M=\mathbb CP^2\#\overline{CP^2}$. This has $H_2(M,\mathbb Z)=\mathbb Z H+\mathbb Z E$ where $H^2=1$ and $E^2=-1$ are the standard generators of the homology of $\...
- 33
5
votes
1
answer
187
views
Second Stiefel-Whitney class of $\mathbb{C}\text{P}^2\#\overline{\mathbb{C}\text{P}^2}$
I know that $w_2(\mathbb{C}\text{P}^2\#\overline{\mathbb{C}\text{P}^2})\neq 0$, where $\overline{\mathbb{C}\text{P}^2}$ is $\mathbb{C}\text{P}^2$ with opposite orientation. But how do you prove this? ...
- 153
2
votes
1
answer
97
views
Can one set and use some axioms for calculating intersection forms of manifolds?
I want to calculate the intersection form of some (four?) manifolds, and I wonder is there any axioms that one can compute the intersection form of (four?) manifolds just by them? like axioms of ...
- 8,425
1
vote
1
answer
130
views
Proof of the Extended Reeb's Theorem for dimensions less than 7
Reeb's Theorem states that every compact smooth manifold which admits a Morse funtion with exactly two critical points is homeomorphic to n-sphere.
I have heard an extension of this theorem for ...
- 25
1
vote
0
answers
171
views
How do I calculate the circumference of a n-sphere?
I have a 4-dimensional sphere of radius r. From it, I can calculate the 3-dimensional surface area of the hypersphere using the formula 2pi^2r^3. I now need to calculate it's circumference, but I'm ...
- 111
0
votes
1
answer
136
views
How many angles it's needed to define 2-plane in 4d?
I mean the angles between arbitary 2-plane and euclidean orthonormal 2-planes which common origin lies at that 2-plane ⊂ R⁴. I think the orthonormality of 2-planes (bivectors) is unambiguous in normal ...
- 11
2
votes
0
answers
73
views
Proof explanation: every 2-sphere in the boundary of a 4-dim handlebody bounds a 3-ball
Suppose that $X$ is 4-dimensional handlebody (meaning a union of 4-dimensional 0 and 1-handles) and $S\subset \partial X$ is an embedded 2-sphere.
The author wants to prove that $S$ bounds an embedded ...
- 685
1
vote
1
answer
346
views
Differentiable structure on $S^4$
How to show that the claim that there is exactly one differentiable structure on $S^4$ implies the smooth four-dimensional Poincaré conjecture (homotopy equivalent to $S^4$ implies diffeomorphic to $S^...
- 33
-1
votes
1
answer
49
views
Combination of reflection symmetries in $\mathbb{E^4}$
Is the combination between point reflection (https://en.wikipedia.org/wiki/Point_reflection) symmetry and hyperplane(or axial using Hodge duality) reflection symmetry (https://en.wikipedia.org/wiki/...
- 293
1
vote
0
answers
57
views
Small exotic $\mathbb{R}^4$'s with symmetries
Definition: An exotic $\mathbb{R}^4$ is a smooth open 4-manifold $E\mathbb{R}^4$ that is homeomorphic to $\mathbb{R}^4$ but not diffeomorphic to it.
Definition: An exotic $E\mathbb{R}^4$ is called ...
- 434
2
votes
0
answers
70
views
Spin connection on Ricci-flat anti-self-dual 4-manifolds
During a talk I heard it was claimed (without proof) that the canonical connection $\nabla^{S^-}$ on the bundle $S^-\to X^4$ of negative chirality spinors over a spin, Ricci-flat, anti-self-dual (ASD) ...
- 767
5
votes
1
answer
147
views
When is the orientable double cover of a product of non-orientable surfaces spin?
Let $M_{k,l}$ denote the orientable double cover of the non-orientable four-manifold $k\mathbb{RP}^2\times l\mathbb{RP}^2$; here $k\mathbb{RP}^2$ denotes the connected sum of $k$ copies of $\mathbb{RP}...
- 96.9k
0
votes
1
answer
137
views
Index of Dirac operator and Chern character of symmetric product twisting bundle
I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text (see image below).
We are twisting the spinor bundle $\Sigma$ with an ...
1
vote
0
answers
14
views
smooth parametrized journey 3d representation of 4d dataset
I have a dataset of $N$ points in 4-D space.
$ \left \{ \left( x_1^{(1)}, x_2^{(1)}, x_3^{(1)}, x_4^{(1)} \right), \left( x_1^{(2)}, x_2^{(2)}, x_3^{(2)}, x_4^{(2)} \right), ... \left( x_1^{(N)}, ...
- 1,020
5
votes
2
answers
173
views
Closed orientable 4-manifolds with $H_2(M)\cong \mathbb{Z}$ do not admit free actions of $\mathbb{Z}/2$
The questions asks us to show that if $M$ is a closed orientable 4-manifold such that $H_2(M)$ is rank $1$, then $M$ does not admit a free action of $\mathbb{Z}/2$.
My attempt has been to suppose $M$ ...
user683708
2
votes
0
answers
152
views
Classification of line bundles over surfaces
I'm currently trying to understand the blow-up process for 4-manifolds. A step in this journey is to understand, topologically, what happens when you pluck out a 4-ball and replace it with $\mathbb{CP}...
- 2,884
6
votes
1
answer
156
views
Example of "practical" applications of Donaldson Invariants
I'm studying Donaldson Invariants from chapter 9 of The wild world of 4-manifolds by Scorpan, and I'm looking for an example where they're used to distinguish two 4-manifolds which are homeomorphic ...
- 2,231