# Questions tagged [4-manifolds]

Questions specifically about $4$-dimensional manifolds

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\mathbb{C}P^2$ 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 ...
1 vote
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### 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 ...
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### 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 ...
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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 $K_0$ ...
1 vote
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### 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 ...
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### 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 ...
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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 ...
1 vote
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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 ...
1 vote
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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 ...
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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