Consider two arcs \(\alpha\) and \(\beta\) embedded in \(D^2 \times I\) as shown in the figure. The loop \(\gamma\) is nullhomotopic in \(D^2 \times I\) (since \(D^2 \times I\) is contractible), but show that there is no nullhomotopy of \(\gamma\) in the complement of \(\alpha \cup \beta\).
Solution
1. Fundamental group presentation of the complement \(X = (D^2 \times I) \setminus (\alpha \cup \beta)\):
1.1. The space \(D^2 \times I\) is homeomorphic to the closed 3-ball \(B^3\).The complement of the two properly embedded tangled arcs \(\alpha, \beta\) deformation retracts onto a 2-dimensional CW complex.
standard collar and Wirtinger presentation for tangles in a 3-ball.
1.2. By the Wirtinger presentation, \(\pi_1(X, x_0)\) is generated by the meridians \(x\) and \(y\) encircling the arcs \(\alpha\) and \(\beta\) respectively:\[ \pi_1(X, x_0) \cong \langle x, y \mid r \rangle, \]where the loop \(\gamma\) represents the commutator word \([\gamma] = [x, y] = x y x^{-1} y^{-1}\).
loop \(\gamma\) winds once around \(\alpha\), then around \(\beta\), and reverses direction.
2. Proof that \([\gamma] \neq 1\) in \(\pi_1(X)\) via a permutation representation:
2.1. Define a homomorphism \(\rho: \pi_1(X) \to S_3\) by sending the generators to transpositions:\[ \rho(x) = (1 \, 2), \qquad \rho(y) = (2 \, 3). \]
2.2. Evaluate the homomorphism on the commutator \([\gamma] = [x, y]\):
\[ \rho([\gamma]) = \rho(x y x^{-1} y^{-1}) = (1 \, 2)(2 \, 3)(1 \, 2)^{-1}(2 \, 3)^{-1} = (1 \, 2)(2 \, 3)(1 \, 2)(2 \, 3) = (1 \, 3 \, 2) \neq \operatorname{id}_{S_3}. \]
cycle multiplication in \(S_3\).
2.3. Since \(\rho([\gamma]) \neq \operatorname{id}_{S_3}\), the element \([\gamma]\) is non-trivial in \(\pi_1(X, x_0)\) (\([\gamma] \neq 1\)).
non-trivial image under a group homomorphism implies non-trivial element.
3. Conclusion: Since \([\gamma] \neq 1 \in \pi_1(X)\), the loop \(\gamma\) is not nullhomotopic in \((D^2 \times I) \setminus (\alpha \cup \beta)\).
1 and 2.
Q.E.D.