Question

Suppose $M$ is a closed 3-manifold, $C\subset M$ is a simple closed curve which represent identity in $\pi_{1}(M)$. Then $C$ bounds an immersed disk in $M$.

My question is:

When does it bound an imbedded disk in $M$?

I don't know about it at all. If you have any reference, please tell me. Thank you!

Answer 1

Suppose that $K$ is a simple closed curve in $M^3$. I'll assume that $M$ is orientable, compact, and without boundary. Let $V$ be a closed regular neighborhood of $K$; so $V \cong S^1 \times D^2$ is a solid torus. Let $X$ be the closure of $M - V$; so $X$ is the exterior of $K$. Let $T = X \cap V$; so $T$ is a two-torus. So $\partial X = \partial V = T$ and $M = X \cup_T V$. Note that $T$ is a two-torus. Let $D \subset V$ be a meridian disk; that is, a disk of the form $\lbrace \mbox{pt} \rbrace \times D^2$.

As Igor indicates, the map $\pi_1(T) \to \pi_1(X)$ induced by inclusion has a kernel if and only if there is a embedded disk in $E \subset X$ with boundary on $T$. If $\partial D$ and $\partial E$ meet once then $K$ bounds a disk in $M$.

To recap: the knot $K$ bounds an embedded disk in $M$ if and only if

1. the map from $\pi_1(T) \to \pi_1(X)$ has kernel and
2. the curve that dies ($\partial E$) meets the meridian $(\partial D$) exactly once.

Answer 2

http://en.wikipedia.org/wiki/Dehn's_lemma

Answer 3

Dehn's Lemma (= Papakyriakopoulos' Theorem) asserts: if C represents 0 in $\pi_1M$ and if C is a simple closed curve, then C bounds an embedded disk.

The assumption on C being a simple closed curve is obviously necessary: multiples of C will not bound embedded disks.

Answer 4

Sorry, I don't know how to use comment (perhaps I can't use it). This is my comment to Sam: where do you use $[K]=0$ in $\pi_1{M}$? I feel that your answer is an explain of Dehn's lemma. I guess that "$[K]=0$" visual meaning is "$K\subset D^3 \subset M$" where $D^3$ is a 3 ball(similar to knot in $S^3$).