List of Fundamental Groups of Common Spaces

# List of Fundamental Groups of Common Spaces

So far we have arrived at the following methods for determining the fundamental group of a space:

**1)**Convex spaces have trivial fundamental groups.

**2)**If $X$ and $Y$ are path connected and homotopically equivalent then their fundamental groups are isomorphic.

**3)**If $X$ and $Y$ are homeomorphic then their fundamental groups are isomorphic.

**4)**The fundamental group of a product of spaces isomorphic to the product of their fundamental groups.

**5)**Apply the Seifert-Van Kampen Theorem.

**6)**If $X$ is a connected compact $2$-manifold then either $X$ is a sphere (which has trivial fundamental group), a connected sum of $m$ tori (which has fundamental group $\langle a_1, b_1, ..., a_m, b_m : a_1b_1a_1^{-1}b_1^{-1}...a_mb_ma_m^{-1}b_m^{-1} = 1 \rangle$), or a connected sum of $n$ projective planes (which has fundamental group $\langle a_1, a_2, ..., a_n : a_1^2a_2^2...a_n^2 = 1 \rangle$.

**7)**Obtain a universal covering space $(\tilde{X}, p)$ of $X$. Then the set of covering transformations, $A(\tilde{X}, p) = A(\tilde{X})$ is isomorphic to the fundamental group of $X$.

We now provide a list of some of the fundamental groups that we have looked at thus far:

## Fundamental Groups of Common Spaces

Space | Image | Fundamental Group | Reasoning |
---|---|---|---|

The Real Line, $\mathbb{R}$, and any Interval | Trivial | The real line (and any interval) is convex and is therefore simply connected. | |

The Closed Disk, $D^2$, and any Open Disk | Trivial | The closed unit disk (and any open disk) is convex and is therefore simply connected. | |

The Real Plane, $\mathbb{R}^2$ | Trivial | The real plane is convex and is therefore simply connected. | |

The Circle | $\mathbb{Z}$ | The real line $\mathbb{R}$ is a covering space of $\mathbb{Z}$ with the covering map $p(x) = (\cos 2\pi x, \sin 2\pi x)$. The set of covering transformations $A(\mathbb{R}, p)$ is homeomorphic to $\mathbb{Z}$. Since $\mathbb{R}$ is a universal cover of $S^1$ we have that the fundamental group of $S^1$ is isomorphic to $\mathbb{Z}$. | |

The Sphere | Trivial | The sphere is convex and is therefore simply connected. | |

The Cylinder | $\mathbb{Z}$ | The cylinder is the topological product $S^1 \times [0, 1]$ and so its fundamental group is isomorphic to $\pi_1(S^1, x) \times \pi_1([0, 1], x) \cong \mathbb{Z} \times \{ 1 \} \cong \mathbb{Z}$. | |

The Mobius Strip | $\mathbb{Z}$ | The circle is a deformation retract of the Mobius strip. Therefore, the Mobius strip has fundamental group isomorphic to $\mathbb{Z}$. | |

The Klein Bottle | $\langle a, b : aba^{-1}b = 1 \rangle$ | Apply the Seifert-Van Kampen Theorem. | |

The Torus, $T^2$ | $\mathbb{Z} \times \mathbb{Z}$ | The torus is the topological product of $S^1 \times S^1$ and so its fundamental group is isomorphic to $\pi_1(S^1, x) \times \pi_1(S^1, x) \cong \mathbb{Z} \times \mathbb{Z}$. | |

Connected Graphs $X$ | $F_n$ where $n = |E(X) - E(T)|$ for any maximal tree $T$ of $X$. | The fundamental group of a connected graph is obtained by taking first finding a maximal tree $T$ of $X$. It will be the free group generated by $|E(X)| - |E(T)|$ elements where $|E(X)|$ is the number of edges in $X$ and $|E(T)|$ is the number of edges in the maximal tree. | |

The Bouquet of $2$ Circles | $F_2$ (Free Group on $2$ Generators) | Apply the Seifert-Van Kampen Theorem. | |

The Bouquet of $n$ Circles | $F_n$ (Free Group on $n$ Generators) | Apply the Seifert-Van Kampen Theorem | |

The Real Projective Line, $\mathbb{P}$ | $\mathbb{Z}$ | The projective line is homeomorphic to the circle via a one-point compactification of the real line. So the fundamental group of the projective line is $\mathbb{P}$. | |

The Real Projective Plane, $\mathbb{P}^2$ | $\mathbb{Z}_2$ | Apply the Seifert-Van Kampen Theorem | |

The Connected Sum of $m$ Tori | The group generated by $a_1, b_1, ..., a_m, b_m$ with the relation $a_1b_1a_1^{-1}b_1^{-1}...a_mb_ma_m^{-1}b_m^{-1} = 1$. | The connected sum of $m$ tori is a connected compact $2$-manifold. | |

The Connected Sum of $n$ Projective Planes | The group generated by $a_1, ..., a_m$ with the relation $a_1^2a_2^2...a_n^2 = 1$. | The connected sum of $n$ projective planes is a connected compact $2$-manifold. |

## Fundamental Groups of Less Common Spaces

Space | Fundamental Group | Reasoning |
---|---|---|

The Real Plane Minus the Origin $\mathbb{R}^2 \setminus \{ (0, 0) \}$ | $\mathbb{Z}$ | Apply the Seifert-Van Kampen Theorem. |

The Sphere Minus a Point | Trivial | Stretch the missing point from the sphere until you get a hole in the sphere. Then continue to stretch the hole around to get a curved open disk and eventually just an open disk. Then the open disk is a deformation retract of this space and hence the fundamental group of a sphere minus a point in trivial. |

The Sphere Minus Two Points | $\mathbb{Z}$ | Stretch both missing points from the sphere to get two holes in the sphere that do not intersect. Pull and stretch along these holes to get a cylinder. Then the cylinder is a deformation retract of this space and hence the fundamental group of a sphere minus two points is $\mathbb{Z}$. |

The Torus Minus a Point | $\mathbb{Z} \times \mathbb{Z}$ | Stretch the missing point from the torus until you get a hole in the torus. Then continue to stretch the hole around. You will eventually get two open washers connected at a point. Shrink the washers to circles. Then the bouquet of two circles is a deformation retract of this space and hence the fundamental group of a torus minus a point is $F_2$. |

$\mathbb{R}^3$ Minus $n$ Skew Lines | $F_{2n}$ (Free Group on $2n$ Generators) | The sphere minus $2n$ points is a deformation retract of this space, and further, the bouquet of $2n$ circles is a deformation retract of this space. Therefore the fundamental group of this space is $F_{2n}$. |

Real Projective $n$ Space, $\mathbb{P}^n$, $n \geq 2$ | $\mathbb{Z}_2$ | $S^n$ is a universal cover of $\mathbb{P}^n$. The group of covering transformations of $S^n$ will be the identity homeomorphism and the homeomorphism which maps points on $S^n$ to their antipodes. This homeomorphism has order $2$ and so the fundamental group of $\mathbb{P}^n$ is $\mathbb{Z}_2$. |