Questions tagged [4-manifolds]

Questions specifically about $4$-dimensional manifolds

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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 ...
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0 votes
2 answers
37 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 $...
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1 vote
1 answer
40 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 ...
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2 votes
1 answer
60 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 ...
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1 vote
0 answers
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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 ...
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  • 433
1 vote
0 answers
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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 ...
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  • 703
7 votes
3 answers
164 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 (...
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0 votes
0 answers
29 views

Approximation Theorem for Cellular Maps

In the paper "Approximating cellular maps by homeomorphisms", Siebenmann proved the following theorem: (Approximation Theorem for Cellular Maps) Let $f: M \rightarrow N$ denote a proper ...
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4 votes
1 answer
80 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 $[\omega]$...
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0 votes
0 answers
68 views

Intersection form of a homology manifold

Recall that for a closed orientable 4-manifold $M$ with fundamental class $M$, there is a intersection form $H^2(M)\times H^2(M)\to \Bbb Z$ defined by $(a,b)\mapsto \langle a\cup b, [M]\rangle$, where ...
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2 votes
1 answer
157 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,\...
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2 votes
0 answers
68 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 $Y$ (...
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0 votes
0 answers
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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 ...
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1 vote
1 answer
93 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 ...
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3 votes
0 answers
108 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 ...
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2 votes
0 answers
50 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 $\Bbb Z_{a_1},\dots,\Bbb Z_{a_k}$...
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2 votes
0 answers
36 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 $c_1^2(L)\geq0$...
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0 votes
0 answers
68 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, ...
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0 votes
0 answers
43 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. ...
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8 votes
1 answer
193 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 ...
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3 votes
0 answers
47 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 $...
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4 votes
2 answers
116 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 ...
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0 answers
54 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 ...
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4 votes
1 answer
164 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 ...
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1 vote
1 answer
46 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}...
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3 votes
1 answer
73 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 $\...
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  • 33
5 votes
1 answer
134 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? ...
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  • 153
2 votes
1 answer
83 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 ...
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1 vote
1 answer
94 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 ...
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1 vote
0 answers
89 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 ...
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  • 111
0 votes
1 answer
106 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 ...
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  • 11
2 votes
0 answers
48 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 ...
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  • 675
1 vote
1 answer
194 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^...
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-1 votes
1 answer
43 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/...
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  • 165
1 vote
0 answers
55 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 ...
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  • 414
2 votes
0 answers
55 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) ...
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  • 737
5 votes
1 answer
133 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}...
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0 votes
1 answer
113 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 ...
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1 vote
0 answers
13 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)}, ...
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5 votes
2 answers
149 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 $M$ ...
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2 votes
0 answers
114 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}...
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  • 2,784
6 votes
1 answer
139 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 ...
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  • 2,185
0 votes
1 answer
36 views

Notion of equivalence for intersection forms

Suppose $Q_X$ and $Q_Y$ are intersection forms of simply connected, smooth, closed 4-manifolds $X$ and $Y$. By Freedman, if $Q_X$ is equivalent to $Q_Y$, then $X$ is homeomorphic to $Y$ (though ...
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1 vote
0 answers
41 views

Connecting the Fibres of a Lefschetz Fibration With Disconnected Fibres.

The following is an exercise set by Chris Wendl in his book Holomorphic Curves in Low Dimensions. I'm fairly new to the subject, and wasn't sure how to approach it, so any help with it would be ...
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1 vote
0 answers
37 views

About natural identifications in knot theory

Let's consider a knot $K$ in a general closed oriented 3-manifold $Y$. And for technical simplicity assume $K$ is rationally nullhomologous, ie, $[K]\in H_1(Y)$ is a torsion element. Now choose a ...
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  • 2,113
9 votes
2 answers
320 views

Examples of 4-manifolds with nontrivial third Stiefel-Whitney class $w_3$.

What are some examples of $4$-manifolds $M$ for which the class $w_3(TM)\in H^3(M;\mathbb{Z}/2)$ is nontrivial? Is there a mapping torus with this property? Motivation: I am wondering whether any ...
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  • 391
1 vote
1 answer
38 views

Twisty cross notation

I read Hatcher's 3-manifold paper earlier this year and I ran into several twisty cross notations which denote manifolds that I understand, but I am not exactly sure what the twisty cross is supposed ...
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4 votes
0 answers
78 views

How to think about exotic differentiable structures in manifolds?

I apologize in advance for the vagueness of this question. It is known that there exist differential manifolds that are homeomorphic but not diffeomorphic to spheres (Milnor), and likewise there are ...
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  • 3,278
2 votes
2 answers
244 views

notation for connected sum $\#^n S^2 \times S^2$

What does the symbol $\#^n S^2 \times S^2$ mean in geometric topology? I know the $\#$ symbol refers to a connected sum. So that we delete a disk from each sphere and sew the two spheres according ...
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  • 23.6k
6 votes
1 answer
708 views

A Hodge dual computation on a $4$-dimensional Riemannian manifold

Let $(M,g)$ be a $4$-dimensional smooth Riemannian manifold. I am trying to understand the following exterior algebra computation: Let $x^1,x^2,x^3,x^4$ be local coordinates on $M$ such that the ...
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