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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questions
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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,...
8
votes
1answer
187 views
Is a gluing of homeomorphic Mazur manifolds diffeomorphic to $S^4$?
A recent paper proves the existence of homeomorphic but not diffeomorphic Mazur manifolds (see also examples of exotic pairs of contractible Stein manifolds).
Let's call them $M_1$ and $M_2$. If we ...
5
votes
1answer
242 views
Example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it
I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x\in M$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is ...
4
votes
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159 views
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 \...
5
votes
1answer
180 views
3-balls with the same boundary in $S^4$ differ up to diffeomorphism
I am looking at this recent paper by Budney and Gabai and I am confused by a certain sentence in it. Theorem 4.7 states that if $\Delta_1$ and $\Delta_2$ are two 3-balls smoothly embedded in $S^4$ ...
9
votes
1answer
387 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
We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...
7
votes
0answers
263 views
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 ...
8
votes
1answer
318 views
Topological mapping class groups of 4-manifolds
It is a classical result of Quinn that for a simply-connected
closed $4$-manifold $X$ the isometries of its intersection form
are in one-to-one correspondence with
$\pi_0 \text{Homeo}(X)$. (Isotopy ...
3
votes
1answer
282 views
Gauss-Bonnet-Chern Theorem [closed]
I am currently doing an undergraduate project about Gauss-Bonnet-Chern Theorem. Is there any particular books/papers regarding the application of the theorem in the theory of general relativity?
5
votes
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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,...
9
votes
1answer
274 views
Genus 2 3-manifolds bounding only $X^4$ with $b_2(X^4)$ big?
The genus of a closed orientable 3-manifold $M^3$ is the minimum genus among all Heegaard splitting surfaces for $M$. Every such 3-manifold bounds a compact 4-manifold. Let $I(M)$ denote the ...
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87 views
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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vote
0answers
101 views
The effect of the Hodge $\star$ operator on the symplectic structure of a Kahler $4$ manifold
Let $(M,\omega, J, g)$ be a $4$ dimensional Kahler manifold. Put $\omega'=\star \omega$ where $\star$ is the Hodge operator associated the metric $g$.
Is $(M,\omega ')$ a symplectic manifold? Is it ...
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votes
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$\exists X$?(No basis for $H_2(X)$ contains spheres)
Does there exist a smooth simply connected closed 4-manifold $X$ with the property below?
Every smooth basis for $H_2(X)$ contains a surface with genus $\geq 1$.
I understand that in general the ...
4
votes
0answers
110 views
(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 6, ...
2
votes
1answer
154 views
Flat scalar curvature on 4 manifold
Let $(M,g)$ be a closed(oriented) Riemannian 4 manifold. It is well-known that, if $scal^g\geq0$ and not identically zero, then $M$ admits a PSC metric by conformal transformation.
Q Is $T^4$ the ...
7
votes
1answer
167 views
ASD connection for Line bundle over $4$-manifold
Let $(M,g)$ be an oriented closed Riemannian $4$ manifold.
Let $L\to M$ be a complex line bundle.
Q Under what condition, we can find an ASD connection of $L$, i.e. a connection $A$ such that $F^+...
4
votes
1answer
149 views
Self-dual differential on $4$-manifold with boundary
Let $(M,g)$ be an oriented compact Riemannian $4$-manifold with boundary $\partial M$.
Let $a\in \Omega^1$ such that $*a\big|_{\partial M}=0$, i.e. $a(\nu)=0$ on $\partial M$, where $\nu$ denotes the ...
8
votes
0answers
267 views
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-...
6
votes
1answer
227 views
Can an exotic diffeomorphism of the 4-ball change the isotopy class of an embedded surface?
Let $W$ be $B^4$ or $S^3 \times I$. Let $Y$ be a properly embedded surface in $W$. Let $f : W \to W$ be a diffeomorphism which is the identity near $\partial W$. Very little is known about $\pi_0(\...
3
votes
1answer
194 views
Obtaining the bounding 4-manifold from the Heegaard diagram
It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the ...
13
votes
2answers
534 views
Example of two exotic closed 4-manifolds s.t. SW(X)=0
I am interested in seeing examples of two closed 4-manifolds $X_1,X_2$ such that $SW(X_i)=0$ and they are homeomorphic but not diffeomorphic.
So far in the literature I've only found examples which ...
4
votes
0answers
125 views
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 ...
3
votes
0answers
236 views
A user guide to the theory on Corks
I am trying to digest the meanings of the corks from the both:
algebraic topology
and
geometry topology
perspectives.
Studying corks is important for understanding the exotic phenomenon of 4-...
3
votes
0answers
76 views
Handlesliding a two component, linking number 1 link
Let $L = K_1 \cup K_2 \subset S^3$ be a two component framed link with $lk(K_1,K_2) = 1$. Let $\hat{L}$ denote the set of all links obtained by handlesliding $L$ around an an arbitrary number of ...
14
votes
0answers
234 views
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-...
13
votes
1answer
356 views
Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent?
Summary
Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the ...
8
votes
1answer
238 views
Relation between the Casson-Gordon invariants $\sigma(M, \chi)$ and $\sigma_r(M, \chi)$
Setting: There are two objects in knot theory that are commonly referred to as the Casson-Gordon invariants: the invariant $\sigma$, and the invariant $\tau$ (see for example A. Conway’s notes ...
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votes
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346 views
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 \\
...
5
votes
0answers
173 views
Homology spheres bounding homology balls but not embedding into $S^4$
Are there any examples of integer homology spheres $Y^3$ that bound smooth integer homology balls but that do not smoothly embeded into $S^4$?
6
votes
1answer
345 views
smooth homotopy 4-balls with sphere boundary in dimension 4
What follows is, as far as I can tell, totally standard folklore. I have one particular point of confusion, other than that, I wanted to confirm that I am uttering the incantations correctly.
The ...
6
votes
1answer
205 views
Framings for 2-surgeries on 4-manifolds
I'm interested in doing $2$-surgeries to $\sharp^k S^1 \times S^3$. That is to the manifold obtained from applying $1$-surgeries to $S^4$.
Since $\pi_1(O(3)) = \mathbb{Z}_2$, there are two possible ...
4
votes
1answer
148 views
Simple invariants to detect concordance in general 3-manifolds
Let $Y$ be a closed, connected, orientable 3-manifold. We call to oriented knots $K_1, K_2$ in $Y$ (smoothly) concordant if there is a smoothly, properly embedded annulus in $Y \times I$ such that ...
2
votes
1answer
223 views
Embedding problem for 3-manifolds attacked via 4-manifolds
In this archiv paper which is continuation of following:
Borodzik, Maciej; Némethi, András; Ranicki, Andrew, Morse theory for manifolds with boundary, Algebr. Geom. Topol. 16, No. 2, 971-1023 (2016). ...
37
votes
1answer
821 views
Exotic $R^4$ as the universal covering space
Is there a smooth compact 4-manifold whose universal covering is an exotic $R^4$, i.e. is homeomorphic but not diffeomorphic to $R^4$?
Remark. I am aware of examples (due to Mike Davis) of compact $...
9
votes
1answer
507 views
Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds?
Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and ...
7
votes
1answer
120 views
Minimum number of double points over all immersed disks
Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi_1(\partial X) \to \pi_1(X)$. Then $K$ is the boundary of some immersed disk $D \to ...
9
votes
3answers
996 views
Is there a closed non-smoothable 4-manifold with zero Euler characteristic?
I will just repeat the title:
Is there a closed non-smoothable 4-manifold with zero Euler
characteristic?
I am guessing yes simply based on other existence theorems I have seen for 4-manifolds.
22
votes
1answer
808 views
What can we say about the Cartesian product of a manifold with its exotic copy?
Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$.
Is it true that $M\times M$ is diffeomorphic to $M\times M^E$?
I am ...
18
votes
2answers
549 views
Behavior of genus function on a 4-manifold for sums
Let $X$ be a smooth compact 4-manifold. Then every element of $H_2(X;\mathbb{Z})$ can be represented by a smooth embedded orientable surface and we have the so called genus function $G: H_2(X; \...
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votes
0answers
86 views
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.
4
votes
1answer
297 views
Canonical Metrics on 3- and 4-Manifolds
From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique ...
6
votes
1answer
480 views
What is “topology in dimension 3.5”?
I've noticed a couple of conference titles which reference something called
"topology in dimension 3.5," such as this one and this one. This subject seems quite mysterious to me — it looks like ...
6
votes
2answers
934 views
Topological Classification of Four-Manifolds
It is known that the topological classification of a closed Riemann surface is determined by its genus. Similar statements are proven for other compact Riemann surfaces with boundaries/marked points. ...
35
votes
3answers
816 views
How to specify a compact topological 4-manifold with a finite amount of data
Finite data for 4-manifolds: I have a somewhat hazy memory of seeing a Ph.D. thesis around 2005 which showed that any compact topological 4-manifold $M$ can be specified by a finite amount of data. ...
4
votes
2answers
449 views
Are there Kirby diagrams with 3-handles?
Let $M\colon \partial_- M \to \partial_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M_i, i \in \{0,\...
2
votes
1answer
145 views
Lefschetz Fibrations and disk bundles
When reading chaptes 7 of Akbulut's book about $4-$manifolds, he describes a handle decomposition for a manifold given a Lefschetz fibration over $S^2$. The idea is to extend the preimage of a disk ...
6
votes
1answer
330 views
Fundamental groups of non-orientable closed four-manifolds
The fundamental group of a closed orientable manifold is finitely presented, and every finitely presented group arises as the fundamental group of a closed orientable four-manifold; see this question. ...
6
votes
2answers
367 views
Decribe the $S^2$ fibration over $S^2$ that gives $\mathbf{CP}^2\#\overline{\mathbf{CP}}^2$
According to this MO post, there is two possible $S^2$ fibration over $S^2$. One is obviously $S^2\times S^2$, another one is the connected sum of two copies of $\mathbf {CP}^2$ with different ...
2
votes
0answers
110 views
Surgery and Curvature on Foliation
Let $X$ be an oriented closed smooth $4$-manifold. Suppose that $TM$ admits a foliation $\mathcal F$ of dimension two, and admits a positvescalar curvature.
Q: If we do the surgery on $X$ to reduce ...
