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Questions tagged [4-manifolds]

Questions specifically about $4$-dimensional manifolds

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Uniqueness of h-cobordisms between 4-manifolds

Kirby in this paper claims (page 4) that "... any two h-cobordisms between $M_0$ and $M_1$ are diffeomorphic" citing two papers of Kreck. In the latter of these papers the main result is the ...
failedentertainment's user avatar
4 votes
1 answer
92 views

Curve of self-intersection $+1$ on complex algebraic surface

Given a compact complex (algebraic) surface, assume it contains a holomorphic curve whose self-intersection equals $+1$. Is there anything we can say about this complex surface? More precisely, will ...
Maths007's user avatar
2 votes
0 answers
69 views

Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

Let $X$ be a compact Riemannian 4-manifold, $P$ a principal $G$-bundle over $X$, and $\mathfrak{g}$ be its adjoint bundle. Let $\omega$ be a self-dual conneciton on $P$ (i.e. its curvature $\Omega \in ...
user302934's user avatar
  • 1,620
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0 answers
28 views

Proof of the fact that every 4-dimensional triangulation is PL

I've been trying to wrap my head around the fact that every triangulation of a 4-manifold must be PL. I have found the following answer: Equivalence of triangulations and piecewise-linear ...
homologic's user avatar
1 vote
1 answer
55 views

Is there any 4-manifold which is known to have a unique smooth structure?

A well-known problem in 4-manifold topology is the smooth Poincaré conjecture, which states whether any smooth manifold homeomorphic to S⁴ is actually diffeomorphic to S⁴, that is, whether S⁴ has a ...
homologic's user avatar
2 votes
1 answer
43 views

A 4-dimensional 3-handle attachment to $S^2 \times D^2$ to get $D^4$.

Given $Y= S^1\times D^2$ and attaching sphere of 3-handle as $S= S^2 \times \{pt\}$, then how to visualise that after attaching 3-handle we get $D^4$ as the resultant manifold? This is part of the ...
Prerak Deep's user avatar
6 votes
2 answers
163 views

Understanding $\mathbb{C}P^2$

I am trying to understand $\mathbb{C}P^2$. Since I understand the Hopf fibration quite well, I like the following construction: Attach a $\mathbb{D}^2$ (2-cell) to a point $\mathbb{D}^0$ (0-cell) to ...
Thomas's user avatar
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3 votes
1 answer
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Showing a 4-manifold is contractible

In Gompf-Stipsicz book, we're presented with the Akbulut cork, and a brief explanation of why it is contractible (see below). Would someone be able to explain what homotopy he is referring too? I ...
horned-sphere's user avatar
3 votes
0 answers
56 views

Intersection Form from a Kirby Diagram

If there are only 2 handles and no 1 handle in a Kirby Diagram then the intersection form of the underlying simply-connected 4-manifold coincides with the linking form. But what if there’s at least ...
horned-sphere's user avatar
0 votes
0 answers
47 views

Intersection between torsion cycles and free cycles

Let's consider a closed and oriented 4-manifold $M_4$ and denote $H_2(M_4,\mathbb{Z})$ as the homology group of 2-cycles and $Q_{M_4}(S_{A},S_{B})$ as the symmetric intersection pairing between 2-...
JQ Skywalker's user avatar
2 votes
1 answer
96 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 ...
Léo S.'s user avatar
  • 356
0 votes
0 answers
67 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 ...
maxsalo12's user avatar
0 votes
1 answer
211 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 ...
CeyhunElmacioglu's user avatar
4 votes
0 answers
60 views

Clarification on constructing a smooth map near singular points in Miller and Piccirillo's proof

I recently asked a question about the proof of theorem 1.8 in Miller and Piccirillo's paper: Knot Traces and Trace Lemma I will continue to keep the same notation as in my previous question. Following ...
amd1234's user avatar
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5 votes
1 answer
212 views

Knot Traces and Trace Lemma

I'm currently reading the paper by Miller and Piccirillo 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 ...
amd1234's user avatar
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5 votes
1 answer
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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 $K_0$ ...
Overflowian's user avatar
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1 vote
1 answer
68 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 ...
Terry Black's user avatar
7 votes
0 answers
220 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 ...
user302934's user avatar
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0 votes
2 answers
161 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 $...
horned-sphere's user avatar
1 vote
1 answer
78 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 ...
user302934's user avatar
  • 1,620
2 votes
1 answer
161 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 ...
Zerox's user avatar
  • 2,054
1 vote
0 answers
51 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 ...
ali_ns's user avatar
  • 503
1 vote
0 answers
50 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 ...
puzzlet's user avatar
  • 785
7 votes
3 answers
177 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 (...
Filippo Bianchi's user avatar
4 votes
1 answer
140 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 $[\omega]$...
Michael Albanese's user avatar
2 votes
1 answer
282 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,\...
blancket's user avatar
  • 1,802
2 votes
0 answers
206 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 $Y$ (...
blancket's user avatar
  • 1,802
0 votes
0 answers
57 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 ...
Darkreaper's user avatar
2 votes
1 answer
223 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 ...
blancket's user avatar
  • 1,802
3 votes
1 answer
171 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 ...
Partha's user avatar
  • 1,419
2 votes
0 answers
69 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 $\Bbb Z_{a_1},\dots,\Bbb Z_{a_k}$...
user302934's user avatar
  • 1,620
2 votes
0 answers
42 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 $c_1^2(L)\geq0$...
blancket's user avatar
  • 1,802
0 votes
0 answers
116 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, ...
Giacomo Bascapè's user avatar
0 votes
0 answers
74 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. ...
cactus314's user avatar
  • 24.5k
9 votes
1 answer
428 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 ...
Michael Albanese's user avatar
3 votes
0 answers
88 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 $...
blancket's user avatar
  • 1,802
4 votes
2 answers
215 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 ...
Christian's user avatar
  • 378
0 votes
0 answers
217 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 ...
Syd's user avatar
  • 1
4 votes
1 answer
425 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 ...
Chris Ewing's user avatar
1 vote
1 answer
79 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}...
Quaere Verum's user avatar
  • 3,213
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 $\...
aeg's user avatar
  • 33
5 votes
1 answer
217 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? ...
Kafka91's user avatar
  • 153
2 votes
1 answer
107 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 ...
C.F.G's user avatar
  • 8,571
1 vote
1 answer
178 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 ...
Frank's user avatar
  • 50
1 vote
0 answers
210 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 ...
Mat NX's user avatar
  • 111
0 votes
1 answer
162 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 ...
Eusa's user avatar
  • 11
2 votes
0 answers
87 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 ...
Giulio's user avatar
  • 705
1 vote
1 answer
461 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^...
Mantu Das's user avatar
-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/...
bonif's user avatar
  • 291
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 ...
melomm's user avatar
  • 454