Notes - Complex Analysis MT23, Homotopies
Flashcards
Suppose:
- $U$ is an open set in $\mathbb C$
- $a, b \in U$
- $\eta : [0, 1] \to U$
- $\gamma : [0, 1] \to U$
- $\eta(0) = \gamma(0) = a$
- $\eta(1) = \gamma(1) = b$
What does it mean for $\gamma$ and $\eta$ to be homotopic?
There exists a continuous function $h : [0, 1] \times [0, 1] \to U$ such that
\[\begin{aligned} &h(0, s) = a\\\\ &h(1, s) = b \end{aligned}\]and
\[\begin{aligned} &h(t, 0) = \gamma(t)\\\\ &h(t, 1) = \eta(t) \end{aligned}\]What is the constant path $c _ a : [0, 1] \to U$?
\[c_a(t) = a\]
for all $t$.
What does it mean for a closed path $\gamma$ starting and ending at $a$ to be null-homotopic?
It is homotopic to the constant path $c _ a$.
Suppose $U$ is a domain in $\mathbb C$. What does it mean for $U$ to be simply-connected?
Any two paths between $a$ and $b$ are homotopic in $U$.
In the context of homotopies, what are convex and star-like sets an example of?
They are simply-connected.