If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values …

There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions …

I have a somewhat general and vague question in mind. Is there anything in literature related to Affine varieties as examples of Stein manifolds? I know that there is a topological …

I have a basic question which I am not able to figure out. If we do a Gluck twist on a nullhomologous 2-sphere in a 4-manifold, it is said that it does not change its intersection …

I'm a novice in the surgery theory, and I'm encountering the word "(4-dimensional) topological surgery conjecture", which I can't not find the definition. Could anyone help me?

In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we kn …

On the first page of Milnor-Kervaire's paper "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", they assert without proof or reference that if $M$ is a compact connecte …

In this nice paper Mikhalkin uses certain (more geometrical than algebraic) aspects of tropical geometry to prove that every complex projective hypersurface in $\mathbb C \mathbb P …

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 th …

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 conseque …

Let us consider a map from a $\Sigma_g \longrightarrow N$, where $N$ is a symplectic manifold. Then we define the moduli space as $M= \{ f | f \mbox{ is a pseudoholomorphic map …

Let $E_{1},E_{2}$ be elliptic curves over $\mathbb{C}$. We denote by $\iota_{i}$ the translation by a 2-torsion point on $E_{i}$. Then $G=\mathbb{Z}/2\mathbb{Z}$ acts freely on the …

[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 …

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 = …

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 c …