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Questions tagged [4-manifolds]

Questions specifically about $4$-dimensional manifolds

In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).

4-manifolds are important in physics because in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.

It can also be used for constructions specific or typical for $4$-manifolds, e.g. the signature, Kirby diagrams, Akbulut diagrams, exotic $\mathbb{R}^4$s, etc.

103 questions
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Are there any simply connected parallelizable 4-manifolds?

On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's confusing to me whether he means that there are no such…
Aru Ray
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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 such $4$-manifolds can be "built out of" a…
aaa
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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 that there could be elliptic surfaces which are…
Michael Albanese
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Pontryagin class of self-dual forms on a 4-manifold

Let $X$ be an oriented Riemannian 4-manifold. The bundle of 2-forms $\wedge^2 X$ can be decomposed into the bundle of self-dual and anti-self-dual forms, $\wedge^2_+ X \oplus \wedge^2_- X$, using the Hodge star. I would like to show that…
Chi Cheuk Tsang
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Uncountable differential structures on $4$-manifolds?

The professor of an introduction to general relativity made a remark that confused me. It is not merely that I find it unintuitive, I also find it hard to wrap my head around what it means: In dimensions $1$ to $3$, a topological manifold can be…
user56834
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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 disk $D$ in $B^4$. Let $Y$ be the 3-manifold…
user302934
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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 (here $F(n)$ denotes the $n$-th Fibonacci number,…
Filippo Bianchi
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Positive scalar curvature in dimension 4

Let $M^n$ be a compact simply connected spin manifold. Gromov, Lawson, and Stolz proved that if $n\geq 5$, then $M$ admits a metric of positive scalar curvature iff $\alpha(M)=0$. Question: What happens in dimension 4? Are there compact simply…
rpf
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non-orientable 4-manifolds

Most of the books and texts I read about classification problems surrounding 4-manifolds which are closed and orientable (with a occasional side-track to open orientable 4-manifolds). This is understandable when you look at it from a physics point…
Willem Noorduin
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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 get $S^2$ (thanks to @Leo Mosher for suggesting…
Thomas
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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 but not diffeomorphic. I know that these…
Bargabbiati
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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 Riemannian volume form of $g$ is…
Asaf Shachar
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Second Stiefel-Whitney class of self-dual two forms of four manifolds

I am trying to understand why for any oriented Riemannian four manifold $X$, we have following equality: $$w_2(X)= w_2(\Lambda^{+}_{2}(TX))$$ Where $\Lambda^{+}_{2}(TX)$ is the bundle of self-dual two forms. I am looking at Milnor-Stasheff's book…
muratguner
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Euler characteristic zero but non-parallelizable four-dimensional manifold

The title speaks for itself. Does there exist a four-dimensional smooth manifold which admits a non-vanishing continuous vector field (Euler characteristic zero) but not a global frame (non-parallelizable)? I've read about five-dimensional examples,…
Bedovlat
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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 following: Theorem: Let $M_0$ and $M_1$ be fixed closed…
failedentertainment
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