# Tagged Questions

The tag has no usage guidance, but it has a tag wiki.

185 views

### A proof of Theorem 9.2.12. in the Gompf-Stipsicz

I'm seeking for a proof of Theorem 9.2.12. in the Gompf-Stipsicz "4-Manifolds and Kirby Calculus" (for the statement, see the following image). But the textbook omits any proofs and only gives a ...
246 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 ...
168 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 ...
167 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 ...
72 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 ...
129 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 $1$?
304 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. ...
195 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 ...
159 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 ...
182 views

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 \# \... 1answer 180 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 ... 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 ... 0answers 592 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 ... 0answers 387 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-... 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 ... 1answer 177 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 ... 1answer 591 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})/\... 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 ... 2answers 255 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 \... 2answers 216 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 \... 1answer 181 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}... 2answers 323 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 ... 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 ... 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 ... 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 ... 1answer 318 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 ... 0answers 125 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 ... 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 ... 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 ... 1answer 158 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) ... 0answers 155 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 S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\} \end{... 1answer 744 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 ... 1answer 314 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 (... 0answers 159 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. ... 2answers 404 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{... 0answers 154 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 ... 1answer 536 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 ... 0answers 322 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 ... 1answer 259 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})\... 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 ... 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-... 0answers 230 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 ... 1answer 388 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 ... 1answer 275 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 ... 2answers 392 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 ... 0answers 198 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 ... 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 ... 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 ...
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 $N$ ...