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3
votes
2answers
245 views

CW 4 manifolds with single 4 cell

Let $M$ be a connected compact closed 4 manifold. Then $H_4(M)=\mathbb{Z}$. If we assume it is smooth, from Morse theory we know that $M$ has a CW structure. But can we find a CW structure of $M$ with ...
4
votes
1answer
163 views

How to understand Taubes' moduli space of holomorphic curves?

Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is ...
4
votes
1answer
163 views

contact surgery diagram on Brieskorn manifolds

For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a ...
3
votes
0answers
71 views

Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
0
votes
2answers
115 views

obtaining circle bundle over torus by trefoil surgery

Does any integer surgery on a right or left trefoil knot give the $S^1$-bundle over $T^2$ with Euler number $1$?
6
votes
1answer
287 views

Construction of invariants of 4-manifolds with the Kirby calculus

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory. I have a question. In the knot theory, the Reidemeister moves play fundamental roles. ...
4
votes
0answers
194 views

Are all 4-manifolds $Pin^{\tilde{c}}$?

It's known that all oriented 4-manifolds admit a $Spin^c$ structure, ie. a spin structure on $TX\oplus\mathcal{L}$ for some complex line bundle $\mathcal{L}$. A usual generalization of this ...
4
votes
1answer
154 views

Homology 3-sphere with a unique Stein-fillable contact structure

Are there any known examples of oriented integer homology 3-spheres $Y$ (besides $S^3$) which have exactly one Stein-fillable contact structure up to isotopy? Failing that, what are the known examples ...
3
votes
1answer
181 views

Simply-connected 4-manifolds can be blown up and down to complex projective planes. How about non-simply-connected ones?

There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$: $$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# \...
8
votes
1answer
175 views

Essential Klein bottle in simply connected symplectic 4 manifolds

Consider the following question: Let $X$ be a simply connected, symplectic 4-manifold. Does there exists a smoothly embedded Klein bottle $K\subset X$ such that the following conditions are both ...
4
votes
0answers
131 views

No irreducible parallelizable manifold of given dimension

What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible? Topological irreducible: it is not homemorphic to ...
18
votes
0answers
565 views

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 $...
12
votes
0answers
386 views

Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Let $K$ be a knot in the 3-sphere $S^3$. Here we denote by $s(K)$ Rasmussen's s-invariant for $K$, and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$ by attaching a $2$-...
23
votes
1answer
1k views

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 ...
3
votes
1answer
176 views

Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise. For any $x \in E$, there is a ...
15
votes
1answer
590 views

Smooth 4-manifolds with $E_8$ intersection form

Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on $H^2(M;\mathbb{Z})/\...
27
votes
3answers
2k views

What manifolds are bounded by RP^odd?

Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that ...
3
votes
2answers
247 views

Legendrian knot in 3-sphere

We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\...
3
votes
2answers
208 views

Brieskorn homology spheres

We know that a Brieskorn homology 3-spheres $\Sigma(p,q,r)$ admit a free $S^1$-action, which makes it a Seifert fibered spaces with three singular fibers: $M(b;r_1,r_2,r_3)$. How should one get from $\...
5
votes
1answer
178 views

Embedded spheres in the K3 surface

Using the Seiberg-Witten theory, we know that every (smoothly) embedded $S^{2}$ in $K3$ with trivial normal bundle is null-homologous. We know we have a lot of interesting knotted $S^{2}$ inside $S^{4}...
9
votes
2answers
321 views

Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$

In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated ...
13
votes
2answers
2k views

What does Yang-Mills and mass gap problem has to do with mathematics?

I'm not very experienced in this topic, but I read a short description of the Yang-Mills existence and mass gap problem, and as long as I understood it has mainly physical consequences and ...
5
votes
2answers
246 views

Disjoint curves in an algebraic surface

Let $X$ be an algebraic surface (over the complex) with $p_g=q=0$. Is it possible to have disjoint curves $C_1,\ldots, C_b$, of positive genus, spanning $H_2(X,{\mathbb Q})$, $b=b_2(X)$? (When $X$ is ...
63
votes
9answers
7k views

Theoretical physics: Why not just R^4?

You and I are having a conversation: "Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles." "Something like that" "...And ...
10
votes
1answer
316 views

Piecewise linear (PL) structures on $\mathbf R^4$

One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference ...
4
votes
0answers
122 views

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 ...
14
votes
4answers
1k views

How Many 4-Manifolds are Symplectic?

As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half (perhaps under some mild assumptions)? I ask this question because ...
4
votes
1answer
230 views

Are “Unions” of small exotic $\mathbb{R}^4$'s small?

Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$. Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup ...
0
votes
1answer
153 views

relations between intersection form and Chern classes

Is there exist a 4-manifold which intersection form has the following property $$ (a,a) \neq 0\ \text{if}\ a\neq 0, $$ and the second (or the first) Chern class (for some almost complex stucture) ...
2
votes
0answers
154 views

Is complex surface in CP(3) a two handlebody?

Consider a complex surface given by homogeneous equation in $\mathbb{C}P^3$. Without loss of generality, take \begin{equation} S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\} \end{...
11
votes
1answer
742 views

Thurston geometries in dimension 4

In the sense of W. Thurston here, there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3. Question: How many different geometries (in the sense of Thurston) do we have in ...
15
votes
1answer
313 views

Shortest Casson tower containing a slice disk for the attaching curve

A Casson tower is obtained as follows: Start with a properly immersed disk in $\mathbb{B}^4$ - a regular neighborhood of such a disk is called a kinky handle. The boundary of the core disk (...
10
votes
0answers
158 views

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 $S^4$. ...
2
votes
2answers
403 views

Approximation theorem for Anti-Self-Dual Metrics

Rounge's Theorem states that any meromorphic function on a domain inside $\mathbb{C}$ can be approximated (over compact subsets) by a sequence of rational functions (meromorphic functions on $\mathbb{...
2
votes
0answers
152 views

Definition of the dual spider number and the formula for the first chern class of the triangle

In the process of trying to understand various maps in Heegaard Floer homology I got stuck on the definition of the dual spider number, which, it seems to me, has a combinatorial definition directly ...
12
votes
1answer
532 views

Handlebody decomposition of an open 4-manifold

Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth ...
2
votes
0answers
319 views

Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many ...
5
votes
1answer
257 views

How to embed genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing nontrivial homology class

As the title says, I want to embed the genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing a nontrivial homology class. I know that $H_2(\mathbb{C}P^2 \# \mathbb{C}P^2; \mathbb{Z})\...
7
votes
1answer
295 views

Are 4-dimensional mapping tori always spin?

We know that all compact orientable manifolds of dimension 3 are spin. In 4 dimensions, $CP^2$ is not spin. I would like to ask if all 4-dimensional compact orientable mapping tori are spin? See ...
4
votes
1answer
143 views

0-homologous surface bounds

Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented 3-...
11
votes
0answers
228 views

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 ...
5
votes
1answer
384 views

What are Kirby diagrams of candidate exotic 4-manifolds?

It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard $S^4$. The same is true for the 4-torus and several other manifolds. Handle ...
10
votes
1answer
274 views

Visualising locally flat embeddings of surfaces in R^4

As far as I understand it follows from the work of M. Freedman that there exist locally flat embeddings of two dimensional surfaces in $\mathbb R^4$ that can not be smoothed in the class of locally ...
5
votes
2answers
389 views

Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?

It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more ...
3
votes
0answers
196 views

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 ...
6
votes
2answers
281 views

Knotted projective planes and fake complex projective space

Paul Melvin gave a talk at Knots in Washington last year in which he asked whether the connected sum of an odd twist-spin of a classical knot and a standard cross-cap embedding of ${\mathbb R}P^2$ is ...
0
votes
0answers
110 views

K3 surface minus finite set

Let $S$ be a complex K3 surface, and $P\subset S$ a finite set of points in $S$. It is known that $$ H^i(S,\mathbb{Z})\cong H^i(S\setminus P,\mathbb{Z}) $$ for $0\le i \le 2$. Then the Euler ...
4
votes
1answer
286 views

Casson invariant and signature

In W. Neumann, J. Wahl, "Casson invariant of links of singularities", Comment. Math. Helv.,1990, Vol. 65, Issue 1, pp 58-78 some connection between the Casson invariant and the signature is ...
8
votes
2answers
924 views

When is the connected sum of manifolds orientation-independent?

Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed? If $N$ ...
3
votes
1answer
323 views

Boundaries of smooth manifolds

If one has a smooth simply connected manifold $M^n$ which we know to bound an $n+1$ dimensional manifold $N$, what can be said about a handle decomposition for one in terms of a handle decomposition ...