-1

How we can show in the sphere or disc,there is no essential simple closed curves ?

In the mapping class groups By Benson farb , definition of essential closed curve is : a closed curve is called essential if it is not homotopic to a point, puncture, or a boundary component."

T566y65tt
  • 3
  • 3
    Did you study algebraic topology before reading that book? If you do, you will know how to prove that these spaces are simply-connected. If you did not study algebraic topology, read, say, the 1st chapter of Hatcher's book and make sure you solve exercises. Then continue reading the "Primer." – Moishe Kohan Jan 17 '22 at 11:35
  • @MoisheKohan . Yes I know . In the Algebraic topology by hatcher $ X$ is simply connected if has trivial fundumental group and path connected. So why $X$ has no essential simple close curves? – T566y65tt Jan 17 '22 at 12:54

1 Answers1

1

First, the sphere and disc are both simply connected.

Second, a path connected space $X$ is simply connected if and only if every continuous function $f : S^1 \to X$ is homotopic to a point; see here.

Putting these together, since a simple closed curve is the image of a continuous function $f : S^1 \to X$, it follows that every simple closed curve is inessential.

Lee Mosher
  • 135,265