Intereting Posts

Prove that the greatest integer function: $\mathbb{R} \rightarrow \mathbb{Z}$ is onto but not $1-1$
a function $f$ is differentiable in $\vec{0}$ if $f \circ \gamma $ is differentiable in 0
Antiderivative of $e^{x^2}$: Correct or fallacy?
What does limit actually mean?
How to show that $f'(x)<2f(x)$
Prove that $\sin x+2x\ge\frac{3x(x+1)}{\pi}$ for all $x\in $
find all self-complementary graphs on five vertices
Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$
Continuation of smooth functions on the bounded domain
Coin with unknown bias flipped N times with N heads, what is p(h)?
Is there only one counter example in $K_5$ for $R(3,3)$?
Suggestions for a learning roadmap for universal algebra?
Find constants of function
$a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle
Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$

In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin structure, etc.).

This raises a couple of questiona:

- Representation of $S^{3}$ as the union of two solid tori
- If $M$ is a nonorientable $3$-manifold, why is $H_1(M, \mathbb{Z})$ infinite?
- How does handle attachment work in Morse Theory
- Prove by elementary methods: the plane cannot be covered by countably many copies of the letter “Y”
- Homotopically trivial $2$-sphere on $3$-manifold
- Exotic Manifolds from the inside

1.Is any compact or non-compact 4-manifold obtainable as a (finite or infinite) handle diagram ?

2.What are the properties needed for a compact or non-compact 4-manifold to be represented as a handle diagram ?

3.What are examples of 4-manifolds with no handle diagram ?

The diagrams can be as complicated as you want (so 0-, 2-, 3-, 4-) handles can be present. I do not know if you can get rid of all the 3-handles in the non-compact case.

This question came forth from the discussion explicit "exotic" charts . I am trying to get help of more people on that, then putting those things in comments (the question of explicit charts of an $\mathbb{E}\mathbb{R}^4$ is another one, albeit interesting in it’s own right).

The question is answered by Bob Gompf by email, see my comment for the main part of his answer.

- Isotopy and homeomorphism
- How to Classify $2$-Plane Bundles over $S^2$?
- What is the importance of the Poincaré conjecture?
- A(nother ignorant) question on phantom maps
- explicit “exotic” charts
- How is PL knot theory related to smooth knot theory?
- If $M$ is a nonorientable $3$-manifold, why is $H_1(M, \mathbb{Z})$ infinite?
- Why is the 3D case so rich?

Weird to answer your own question, but one becomes wiser with years. Seens that every 4-manifold can be represented as a Kirby Diagram. Problem is that these things can get very complicated (infinite many 1- or 3-handles, or infinite 0-handles, kinks in the handles, etc). So the question can be answered negatively: there are none.

- Group theory exercise: proof that $b^2 = e$ if $|a|$ is odd and $b = a^{-1}b^{-1}a$
- Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
- Choices of decompositions of a representation into irreducible components (Serre, Ex. 2.8)
- Introductory Calculus textbook for physics?
- What is the use of hyperreal numbers?
- Pointwise limit of a measurable function is still measurable, for weak star convergence measure
- Is there a function such that $f(f(n)) = 2^n$?
- Density of irrationals
- Nice way to express the radical $\sqrt{4+\sqrt{4+\sqrt{4+\dots}}}$
- Constructing the midpoint of a segment by compass
- Locating the quadrant containing a point on an n-sphere
- Set and its Complement are Measure Dense
- Convergence in distribution to derive the expectation convergence
- How can the Gödel sentence be Pi_1
- If $A$ is positive definite then so is $A^k$