Questions tagged [4-manifolds]
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.
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What is the three-dimensional hyperbolic volume of a four-manifold?
Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.
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When is a compact topological 4-manifold a CW complex?
Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres".
Kirby showed that ...
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Exotic 4-spheres and the Tate-Shafarevich Group
The title is a talk given by Sir M. Atiyah in a conference with the following abstract:
I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
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Concordance and homology cobordism
If two knots $K_1$ and $K_2$ in $S^3$ are smoothly concordant, then for any rational number $r$, the $r$-surgeries $S^3_r(K_1)$ and $S^3_r(K_2)$ are homology cobordant. Is the converse true? What if $...
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Which manifolds decompose into pants?
In this nice paper Mikhalkin uses certain (more geometrical than algebraic) aspects of tropical geometry to prove that every complex projective hypersurface in $\mathbb C \mathbb P ^n$ decomposes as a ...
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Almost complex 4-manifolds with a "holomorphic" vector field
Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
The following sub question is ...
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Are there two non-equivalent exotic structures on $\mathbb{R}^4$ coming from topologically slice, non-slice knots?
For a knot $K \subset S^3$, which is topologically slice but not slice (in a smooth way), there's a four manifold $\mathbb{R}^4_K$, homeomorphic but not diffeomorphic to standard euclidean $\mathbb{R}^...
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Does there exist a closed 4-manifold whose $\pi_2$ contains torsion elements
Does there exist a closed 4-manifold $X$ such that $\pi_2(X)$ contains torsion elements?
And, if so, does there exist a closed 4-manifold $X$ such that $\pi_2(X)\neq 0$ but $\pi_2(X)\otimes \mathbb{Q}=...
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Are there exotic twisted doubles of 4-manifolds?
Take a smooth 4-manifold $X$ whose boundary has a diffeomorphism $\tau: \partial X \to \partial X$ that extends to a homeomorphism but not a diffeomorphism of $X$. (By Matveyev and Curtis-Freedman-...
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Nonsmoothable 4-manifolds
Does there exist a closed connected nonsmoothable 4-manifold $M$ such that:
$\kappa(M)=0$ (Kirby-Siebenmann invariant vanishes, hence, there is no "classical" obstruction to smoothability) ...
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Is the Lipschitz structure on $\mathbb{S}^4$ unique?
Sullivan [S] proved that on any $n$-dimensional, $n\neq 4$, topological manifold there is a unique Lipschitz structure. Although it is generally accepted result, it seems that the proof lacks some ...
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Excluding exotic PL structures on S^4
Suppose you have a finite group $G<SO(5)$ such that $S^4/G$ is homeomorphic to $S^4$ and such that $S^4/G$ is a PL manifold with respect to a PL structure induced by a standard structure on $S^4$. ...
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Fox re-imbedding theorem in dimension four
Fox re-imbedding theorem states the following:
A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...
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Isotopy on embedded 3-manifolds in 4-manifolds
Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$...
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Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2
Fix $x,y,z\in \mathbb{C}^*$ and let $M=S^1\times S^1\times S^1$ with $\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$ mapping the three generators to diagonal matrices with entries $(x,x^{-1})$, $(y,...
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Configuration space of 4 points as an orbifold
Setup: Consider the braid group $B_n$. One way to define this is as the fundamental group of the unordered configuration space $UC_n(\mathbb{C}) = \{\{z_1,\dotsc,z_n\}\subset \mathbb{C} \mid z_i \not= ...
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A conjecture about homotopy $S^1\times B^3$'s
$\textbf{Conjecture}:$
Let $X^4$ be a homotopy $S^1\times B^3$ with the following properties:
Attaching a four dimensional 2-handle gives a standard $B^4$.
The $k$-fold cyclic cover is diffeomorphic ...
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Integer surgeries along links yielding lens spaces
Does there exist an integer $N$ such that any lens space $L(p,q)$ can be obtained by integer surgery from $S^3$ along a link $L$ with at most $N$ components?
EDIT:
I have worked out the comment by ...
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Self diffeomorphism of $S^2\times S^2$
The main question is motivated from the answer of this question https://math.stackexchange.com/questions/2481200/finite-groups-gs-which-acts-freely-on-s2-times-s2
Is it true that every self ...
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Lusternik-Schnirelmann Category of 4-Manifolds
Whilst reading 'the book' I stumbled across the following elegant little theorem, due to Gómez-Larrañaga and González-Acuña.
Let $M$ be a closed $3$-dimensional manifold. Then its Lusternik-...
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On exotic symplectic structures of smooth closed 4-manifolds
What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...
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Applications of E8 manifold
The $E_8$ Cartan matrix is given by,
$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
...
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Intersection form of logarithmic transformations
Now I want to calculate the intersection form of a logarithmic transformation which is defined as follows.
Let $X$ be an oriented, closed, simply-connected 4-manifold and $T^2\subset X$
be an ...
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Kirby diagram of Enriques surface (as the "(1/2) K3 surface")
Not to be confused with $E(1)\cong\mathbb{C}P^2\#9\overline{\mathbb{C}P^2}$, which is also known as a $\frac{1}{2}K3$ surface (in the sense that removing a neighbourhood of a regular torus fiber in $E(...
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11/8-type inequality from Heegaard Floer theory?
Background: a well-known conjecture in $4$-dimensional topology states that if $M$ is a $4$-dimensional oriented closed spin smooth manifold, then the second Betti number of $M$ is bounded from below ...
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A generalized Dirac operator
Let $(M^4,g)$ be a closed four-dimensional Riemannian manifold and $J$ be an almost complex structure on $M$. Then for normal coordinate $e_1,\dots e_4$ at a point $m,$ and for a section $\alpha$ of a ...
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Questions about a paper by Laudenbach and Poénaru
I am working on the 1972 paper A Note on 4-Dimensional Handlebodies by F. Laudenbach and V. Poénaru, and I had two questions. I will use their notations to simplify things, since the paper is very ...
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Quotients of 4-sphere by smooth $Z_p$ actions with knotted fixed point sets
This question is closely related to another I asked today.
Giffen showed in 1966 that the generalized Smith conjecture is false by constructing for odd $p$ a smooth $Z_p$ action on $S^4$ with fixed-...
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What is the state of the art in 4-manfold 2-types?
In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...
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Surgering locally flat tori in 4-manifolds
Is there a locally flat torus in some not smoothable topological 4-manifold such that surgering on it produces a smoothable 4-manifold? Surgering means removing a tubular neighborhood and reattaching ...
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Can every homeomorphism of the 4-ball fixing the boundary be approximated by diffeomorphisms smoothly isotopic to the identity?
Let $B^{n}$ denote the euclidean closed ball of dimension $n$. Alexander's trick shows that every homeomorphism of $B^{n}$ fixing $\partial B^n$ is $C^{0}$-isotopic to the identity via a boundary ...
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Standard 2-instantons on the 4-sphere under conformal transformation
It is well-known that there is a standard SU(2) 1-instanton on the 4-sphere and any 1-instanton can be obtained from the standard one by conformal transformation. Is there any explicit (family of) &...
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Curvature of the line bundle $\mathcal{O}(2)$ on the twistor space
Let $M^4$ be a closed Riemannian manifold and $Z:=S\big(\Lambda^2_+(M)\big)$ denote the twistor space of $M,$ i.e., the sphere bundle of the self-dual 2-forms on $M$. Now at a point $(m,J)\in Z$ the ...
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Are there connected closed 4-manifolds admitting a regular Almost Lagrangian distribution, and which are not Lorentzian?
In the category of real differential manifolds, connected (of $ C ^ {\infty} $ class in the sequel), closed of dimension 4, is there any manifold admitting a regular Almost Lagrangian distribution and ...
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Generators of the automorphism group of a quadratic form
Suppose that $q$ is an integral quadratic form, not necessarily unimodular or even non-degenerate, and write $O_q(\mathbb{Z})$ for its automorphism group. According to this question this group is ...
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Homotopy type of a $4$-manifold with finite fundamental group (paper by S. Bauer)
I'm studying Stefan Bauer's paper
The homotopy type of a 4-manifold with finite fundamental group. In: tom Dieck T. (eds) Algebraic Topology and Transformation Groups. Lecture Notes in Mathematics,...
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Candidates of nonsmoothable homology 4-spheres
I'm not trying to duplicate a question Are there non-smoothable homotopy/homology spheres?
But in Igor Belegradek's answer, he pointed out that "Some of them (homology spheres) have large ...
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Obstruction of smooth structure
The first 24 lectures of Jacob Lurie on Geometric Topology [1]
gave a concise introduction to the comparison of smooth manifolds
and piecewise-linear manifold. In the first five lectures, it is
shown ...
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Submanifolds of $4$-manifolds and their intersections
Suppose we have two (oriented) submanifolds $A,B$ of an oriented $4$-dimensional manifold $M$, that intersect transversally. Looking at the standard references for $4$-manifolds, I couldn't find a ...
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An Akbulut cork with a simple equation?
Is there an Akbulut cork that is diffeomorphic to a complex affine surface that can be given by a simple equation? (for example a surface given by zeros of a low-degree polynomial in $\mathbb C^3$)
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Surgery diagrams of 3-dim abstract open books
Starting with an abstract open book $(\Sigma,\phi)$, I would like to understand some of the manifolds that I could obtain. Given a surface $\Sigma$ and monodromy $\phi$, it is not hard to find a Kirby ...
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Involutions on $D^4$ with a fixed arc
By a theorem of Livesay, the 3-sphere has a unique (up to equivariant diffeomorphism) smooth involution with exactly two fixed points. Thinking of $S^3$ as the unit sphere in $\mathbb{R}^4$, this ...
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Smoothability of open 4-manifolds
F. Quinn proved that any open topological 4-manifold admits a smooth structure in Ends of maps III: dimensions 4 and 5.
He first proves the generalized annulus conjecture:
Suppose $h:D^j\times \...
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A 4-manifold with a special non-free circle action?
Let $X$ be an oriented closed 4 manifold, with a nontrivial orientation-preserving circle action.
Question Is there an example such that $X/S^1$ is an orbifold (not a manifold), with a trivial first ...
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(Non-)Orientability of non-triangulable manifolds
We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least ...
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Are triangulations with common refinements PL-homeomorphic?
Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...
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Blow-up and Blow-down kirby local moves for non-orientable $3$-manifold
Can anyone explain or give a reference about the Blow-up and Blow-down Kirby local moves for non-orientable $3$-manifolds?
Thanks, advance.
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Blow-up of $\mathbb{P}^4$ along a smooth surface
Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$).
In dimension ...
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Compact self-dual Einstein metrics of negative scalar curvature
Does a compact four-dimensional self-dual Einstein manifold with negative scalar curvature have negative sectional curvature?
This would be true if we believe the folklore conjecture that a compact ...
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Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s
Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...