what is the meaning of a curve $C$ representing Identity in fundamental group? 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!
 A: 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 


*

*the map from $\pi_1(T) \to \pi_1(X)$ has kernel and 

*the curve that dies ($\partial E$) meets the meridian $(\partial D$) exactly once.

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