2
votes
1answer
49 views

Stability of the nonempty intersection of an open set $A$ with a set $S$ under homotopy?

To be more precise: let $F(x,t) : R^2 \times I \to R^2$ be a homotopy of open maps $F(_,t)(x)$ (the restriction of $F$ to some fixed $t$) (the homotopy is continuous in both variables). Suppose that ...
3
votes
1answer
61 views

Why not extending to the whole disk implies have a zero

For any complex polynomial $p(z)$ of order $m$, we showed earlier that on a circle $S$ of sufficiently large radius $r$ in the plane, $$\frac{p(z)}{|p(z)|}\quad \text{and}\quad ...
1
vote
1answer
134 views

uniformization theorem - squares and circles

I am trying to understand the uniformization theorem and get some intuition about it. The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of ...
7
votes
1answer
179 views

Is manifold mapping degree equal to algebraic degree for polynomials?

If $M$ and $N$ are oriented $n$-manifolds and $f: M \to N$ then the degree of $f$ is given by $$ \deg f = \sum_{p \in f^{-1}(q)} sign_p f $$ where $q$ is a regular value and the sign is $+1$ if $f$ is ...
3
votes
1answer
128 views

smoothness of hopf fibration projection with respect to standard differentiable structure on unit sphere

We know that steographic projection defines a differentiable structure on $S^n$ by sending points on $S^n$ to hyperplane $\{x^{n+1}\}=0$. In fact, stereographic projection $\sigma_P: S^n- P\to R^n $ ...
1
vote
1answer
86 views

smooth domain in $\mathbb{C}^2$ and smooth bounday of bounded domain in $\mathbb{C}^2$

How can we define a smooth domain in $\mathbb{C}^2$ and a smooth boundary of bounded domain in $\mathbb{C}^2$? {where $\mathbb{C}^2$ := Cartesian product of complex plane }
5
votes
1answer
671 views

The winding number and index of curve

If $\gamma $ is a smooth closed curve in $\mathbb R^2-\{0\}$, I want to know whether the winding number of $\gamma $ about $0$, i.e., $\frac{1}{{2\pi i}}\int\limits_\gamma {\frac{1}{z}} dz$ is equal ...
4
votes
1answer
133 views

Holomorphic immersion

Is it possible to get an holomorphic immersion form $\mathbb{C}$ to $\hat{\mathbb{C}}$ which is surjective? Here immersion means that $f'(z)\neq 0$ when $f(z)$ is finite and ${\left(\frac ...
4
votes
1answer
258 views

Lifting of a tangent bundle

I have a problem with Kuranishi's theorem in deformation theory. I'll try to formulate it in general terms, and then describe the particular situation. Let $\pi : M \to S$ be a smooth fiber bundle - ...
2
votes
1answer
187 views

Which smooth 1-manifolds can be represented by a single smooth parametrization?

Among the smooth 1-manifolds (with or without boundary) which embed into $\mathbb{R}^2$, which ones can be represented by a single parametrization $z = (x,y) = f(t)$, for $t \in I$, where $I$ is an ...