# 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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### 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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### 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 ...
1 vote
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### 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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### 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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### 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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### $c_1^2(L)\geq 0$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
1 vote
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### 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
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### 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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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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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### 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 ...
1 vote
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### 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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### 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 ...
### 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 ...
### 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 ...