Write down the fundamental group of the following spaces:
\(\mathbb{R}^2 - \{0, 1\}\)
\(\mathbb{R}^2 - I\)
The symbol \(\oplus \in \mathbb{R}^2\)
\(S^2 - \{p_i\}_{i=1}^4\)
\(T - \{p_0\}\)
\(S^2 / \mathbb{Z}_2\) via the antipodal map
\(S^2/\mathbb{Z}_3\) via a \(2\pi/3\) rotation about the \(z\)-axis.
\(S_2 \cup \{(0,0,z) \mid -1 \leq z \leq 1 \}\)
\(\mathbb{R}^3 - \{ (x,y,0) \mid x^2 + y^2 = 1\}\)
\(\mathbb{R}^2 - H\), the Hopf link
Solution
1. Computation of fundamental groups for items 1 through 5:
1.1. Item 1: \(\mathbb{R}^2 \setminus \{0, 1\}\)The plane minus 2 points deformation retracts onto a figure-eight space \(S^1 \vee S^1\).Thus \(\pi_1(\mathbb{R}^2 \setminus \{0, 1\}) \cong F_2 = \mathbb{Z} * \mathbb{Z}\).
deformation retraction of \(\mathbb{R}^2 \setminus \{2 \text{ points}\}\) onto wedge of 2 circles.
1.2. Item 2: \(\mathbb{R}^2 \setminus I\) (where \(I = [0, 1]\))The closed interval \(I\) is contractible and can be deformed to a single point \(\{0\}\), so \(\mathbb{R}^2 \setminus I \simeq \mathbb{R}^2 \setminus \{0\} \simeq S^1\).Thus \(\pi_1(\mathbb{R}^2 \setminus I) \cong \mathbb{Z}\).
homotopy equivalence of plane minus a segment with punctured plane.
1.3. Item 3: The symbol \(\oplus \subset \mathbb{R}^2\)The space \(\oplus\) is a 1-dimensional CW complex (graph) consisting of 5 vertices (center \(+ 4\) on circle) and 8 edges (4 circle arcs \(+ 4\) spokes).Its Euler characteristic is \(\chi = 5 - 8 = -3\).The fundamental group of any connected graph is free of rank \(1 - \chi = 1 - (-3) = 4\).Thus \(\pi_1(\oplus) \cong F_4\).
fundamental group of finite connected graph.
1.4. Item 4: \(S^2 \setminus \{p_1, p_2, p_3, p_4\}\)Stereographic projection from \(p_4\) gives a homeomorphism \(S^2 \setminus \{p_1, p_2, p_3, p_4\} \cong \mathbb{R}^2 \setminus \{3 \text{ points}\} \simeq \bigvee^3 S^1\).Thus \(\pi_1(S^2 \setminus \{p_i\}_{i=1}^4) \cong F_3 = \mathbb{Z} * \mathbb{Z} * \mathbb{Z}\).
stereographic projection and deformation retraction.
1.5. Item 5: \(T \setminus \{p_0\}\) (punctured torus)Removing a point from the standard 2-cell representation \(T = I^2 / \sim\) deformation retracts the punctured 2-cell onto the 1-skeleton \(S^1 \vee S^1\).Thus \(\pi_1(T \setminus \{p_0\}) \cong F_2 = \mathbb{Z} * \mathbb{Z}\).
deformation retraction onto 1-skeleton.
2. Computation of fundamental groups for items 6 through 10:
2.1. Item 6: \(S^2 / \mathbb{Z}_2\) via the antipodal mapThe quotient of \(S^2\) under the antipodal identification is the real projective plane \(\mathbb{RP}^2\), which has universal cover \(S^2\).Thus \(\pi_1(S^2 / \mathbb{Z}_2) \cong \mathbb{Z}_2\).
covering space classification for \(\mathbb{RP}^2\).
2.2. Item 7: \(S^2 / \mathbb{Z}_3\) via rotation by \(2\pi/3\) about the \(z\)-axisViewing \(S^2 \cong \Sigma S^1\), the \(\mathbb{Z}_3\)-action rotates each latitude circle and fixes the north and south poles.The quotient space is homeomorphic to the suspension \(\Sigma(S^1 / \mathbb{Z}_3) \cong \Sigma S^1 \cong S^2\).Because \(S^2\) is simply connected, \(\pi_1(S^2 / \mathbb{Z}_3) \cong 0\).
suspension of a circle is homeomorphic to \(S^2\).
2.3. Item 8: \(S^2 \cup \{(0, 0, z) \mid -1 \le z \le 1\}\)The space consists of \(S^2\) with an interior diameter segment joining the poles.Collapsing the contractible diameter segment to a point yields the wedge sum \(S^2 \vee S^1\).By the Seifert–van Kampen Theorem:\[ \pi_1(S^2 \cup \text{segment}) \cong \pi_1(S^2 \vee S^1) \cong \pi_1(S^2) * \pi_1(S^1) \cong 0 * \mathbb{Z} \cong \mathbb{Z}. \]
quotient by contractible subcomplex.
2.4. Item 9: \(\mathbb{R}^3 \setminus \{(x, y, 0) \mid x^2 + y^2 = 1\}\)The complement of the unknot in \(\mathbb{R}^3\) deformation retracts onto \(S^2 \vee S^1\) (the \(z\)-axis plus a 2-sphere enclosing the circle), so:\[ \pi_1(\mathbb{R}^3 \setminus S^1) \cong \pi_1(S^2 \vee S^1) \cong \mathbb{Z}. \]
deformation retraction of unknot complement in \(\mathbb{R}^3\).
2.5. Item 10: \(\mathbb{R}^3 \setminus H\) (Hopf link complement)The complement of the Hopf link in \(S^3\) is homeomorphic to \(T^2 \times (0, 1) \simeq S^1 \times S^1\).Thus:\[ \pi_1(\mathbb{R}^3 \setminus H) \cong \pi_1(S^1 \times S^1) \cong \mathbb{Z} \oplus \mathbb{Z}. \]
Wirtinger presentation / torus fiber structure of the Hopf link complement.
3. Conclusion:The fundamental groups are:1. \(F_2\), 2. \(\mathbb{Z}\), 3. \(F_4\), 4. \(F_3\), 5. \(F_2\), 6. \(\mathbb{Z}_2\), 7. \(0\), 8. \(\mathbb{Z}\), 9. \(\mathbb{Z}\), 10. \(\mathbb{Z} \oplus \mathbb{Z}\). Q.E.D.
1 and 2.