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1
vote
2answers
274 views

Group action on spin^c 4-manifold.

[edit] I'll try to be more precise. In paper N.Nakamura, "Bauer–Furuta invariants under $Z_2$-actions" there is an assumption that $Z_2$ action "lifts to spin^c structure". What i think it means: ...
3
votes
1answer
590 views

Homeomorphism classification of 4-manifolds

Question 1. Let $X_i$ be an infinite family of closed, orientable, smooth 4-manifolds with the following properties: a) $\pi_1(X_i) = \mathbb{Z}\times \mathbb{Z_{2}}$ for any $i = 1, 2, \cdots $ b) ...
6
votes
0answers
143 views

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 ...
12
votes
1answer
494 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 ...
8
votes
2answers
838 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 $N$ ...
5
votes
5answers
1k views

What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?

My question is as stated in the title: What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ? I am particularly interested to know if ...
9
votes
1answer
399 views

Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively

I'm interested in a complexity question related to problems like the slice-ribbon problem. To be specific, if $K \subset S^3$ is a knot, it might be non-trivial yet still bound a smoothly-embedded ...
11
votes
2answers
541 views

Existing proofs of Rokhlin's theorem for PL manifolds

I'm looking for a comprehensive reference to existing proofs of Rokhlin's theorem that a 4-dimensional closed spin PL manifold has signature divisible by 16. I'm specifically interested in direct ...
11
votes
1answer
544 views

Smooth four-manifolds with contractible universal cover

Let $X$ be a smooth compact four-manifold with definite non-trivial intersection form. Can the universal cover of $X$ be contractible? It semms to me that the answer is negative when $X$ is simply ...
4
votes
3answers
963 views

Minimal genus, adjunction inequality

Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$. As I know ...
22
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 ...
11
votes
1answer
1k views

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? [closed]

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$?
14
votes
1answer
652 views

Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?

Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper. ...
10
votes
2answers
425 views

Embedding the product of three circles in the 4-sphere.

Let $M$ be a smooth submanifold of the 4-sphere $S^4$. I'm going to demand that $M$ be diffeomorphic to $S^1 \times S^1 \times S^1$. By Jordan-Brouwer separation, $M$ separates the 4-sphere into ...
12
votes
2answers
622 views

Explicit embeddings of Cappell-Shaneson knots

In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare ...
28
votes
0answers
1k views

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. ...
5
votes
2answers
666 views

First appearance of Novikov's additivity theorem

Hi! Novikov's additivity theorem states that if you glue together two compact oriented 4n-manifolds along a connected component of their boundaries, the signature of the resulting manifold is ...
0
votes
0answers
251 views

3-handle cancellation of 4-dimensional handlebody.

Let $X^4$ be the 4-dimensional handlebody with $\partial X=S^3$ and $\pi_i(X)=\pi_i(B^4)$. Is it true that we can always change $X^4$ with handlebody without 3-handle? (I'm concerning about the ...
12
votes
1answer
1k views

A question on classification of almost complex structures on $4$-manifolds

I have a (basic?) question in topology. Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by ...
24
votes
1answer
1k views

Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$ Background Definition (Exotic $\mathbb{R}^4$): An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to ...
69
votes
4answers
6k views

How can there be topological 4-manifolds with no differentiable structure?

This is a very naive question, and I'm hoping that it will be matched by a correspondingly elementary answer. It is well known that not every topological 4-manifold admits a smooth structure. So ...
17
votes
1answer
856 views

What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$? Background Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...
7
votes
1answer
558 views

Are there symplectic 4-folds with $b_+>1$, $b_-=0$?

This is the question. Is it known that a symplectic $4$-fold with $b_2>1$ should have a homology class $C$ with $C^2<0$?
7
votes
1answer
654 views

Gromov-Witten invariants counting curves passing through two points

Let us say that a closed symplectic manifold $X$ is $GW_g$-connected if there is a nonvanishing Gromov-Witten invariant of the form $GW_{g,n}^{X,A}(\beta,point, point,\alpha_3,\ldots,\alpha_n)$ --in ...
13
votes
1answer
1k views

Proofs of Rohlin's theorem (an oriented 4-manifold with zero signature bounds a 5-manifold)

A celebrated theorem of Rohlin states the following An oriented closed 4-manifold $M^4$ bounds an oriented 5-manifold if and only if the signature of $M^4$ is zero. Simple homological arguments ...
2
votes
2answers
367 views

Reference for the proof of this statement?

Can anyone give me the reference for this statement?: Let $M$ be a closed oriented smooth 4-manifold. Any element $a\in H_2(M)$ can be represented by a smoothly embedded, oriented surface. I found ...
22
votes
4answers
2k views

Can all n-manifolds be obtained by gluing finitely many blocks?

Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...
8
votes
2answers
1k views

Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read "If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in possession of a knot which is ...
3
votes
4answers
439 views

Compactify S^2\times S^2-\Delta

Let $\Delta\subset S^2\times S^2$ be the diagonal. Then $S^2\times S^2\setminus \Delta$ is an open four dimensional manifold. By a compactification of it, I mean a closed four dimensional manifold $X$ ...
13
votes
1answer
387 views

Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree?

A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an ...
21
votes
3answers
1k views

How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms. [closed]

With geometrization, Rubinstein's 3-sphere recognition algorithm and the Manning algorithm, 3-manifold theory has reached a certain maturity where many questions are "readily" answerable about ...
9
votes
3answers
706 views

Alexander polynomial or Reidemeister torsion for knotted surfaces?

An important invariant of a knot in $S^3$ is its Alexander polynomial, related also to Reidemeister torsion. Is there something like that for knotted surfaces in $S^4$? If not, what are the ...
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 ...
5
votes
3answers
350 views

PD3 groups and PD4 complexes

I am interested at the moment in what groups can occur as the fundamental group of a 4-manifold (or more generally, a 4-dimensional CW complex) with prescribed conditions on the intersection form. I ...
14
votes
2answers
1k views

topological “milnor's conjecture” on torus knots.

Here's a question that has come up in a couple of talks that I have given recently. The 'classical' way to show that there is a knot $K$ that is locally-flat slice in the 4-ball but not smoothly ...
7
votes
2answers
590 views

slice=ribbon generalization to higher genus + potential counterexamples to slice=ribbon.

I have two questions about the slice=ribbon conjecture. (1) If a knot $K \hookrightarrow S^3$ has smooth slice genus $g$, you can ask if it bounds a smooth genus $g$ surface in $S^3 \times [0, ...
4
votes
4answers
644 views

4-genus of a 2-bridge link

How can we calculate the 4-genus of a link L? The 4-genus is defined to be the minimal genus of orientable surface bounded by L in B^4. Is there any routine method to calculate that? Especially, any ...
0
votes
1answer
258 views

If the 4-genus of a link is zero, is it a slice link?

An n-component slice link is a link that bounds n disjoint discs in B^4. And the 4-genus of a link is defined to be the minimal genus of orientable surfaces bounded by it in B^4. My question is: if ...
18
votes
0answers
693 views

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 ...
24
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 ...
26
votes
3answers
1k views

Embeddings of $S^2$ in $\mathbb{CP}^2$

Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line? Note: I suspect ...
6
votes
2answers
799 views

A problem/conjecture related to 4-manifolds that deserves a name. What name does it deserve?

There's an old problem in 4-manifold theory that, as far as I know, doesn't have a name associated with it and really deserves a name. Let $M$ be a smooth 4-manifold with boundary. Let $S$ be a ...