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The fundamental groups of a few other familiar topological spaces are given by:

The fundamental groups of a few other familiar topological spaces are given by:

* Torus: <math>\pi_1(S^1 \times S^1) \cong \mathbb Z \oplus \mathbb Z</math>.

* Torus: <math>\pi_1(S^1 \times S^1) \cong \mathbb Z \oplus \mathbb Z</math>.  Let <math>a</math> be a loop going around the central hole, and <math>b</math> be a loop around the core of the torus.  It's easy to check that <math>ab</math> is actually homotopic to <math>ba</math>, which means that to specific a loop on the torus, one needs only say how many times it goes around each of these, by specifying an ordered pair <math>(m,n)</math> of integers.

* Mobius strip: <math>\pi_1(M) \cong \mathbb Z</math>.

* Mobius strip: <math>\pi_1(M) \cong \mathbb Z</math>.  As in the circle, to specify a path on the Mobius strip up to homotopy, it suffices to say how many times the path goes around the hole.

* Cylinder: <math>\pi_1(S^1 \times \mathbb R) \cong \mathbb Z</math>.

* Cylinder: <math>\pi_1(S^1 \times \mathbb R) \cong \mathbb Z</math>.  This too works the same way as the circle: a homotopy class of paths is determined by the number of times it wraps around the hole.

* The complement of a trefoil [[knot]] in 3-dimensional space: <math>\pi_1(S^3 \setminus K)</math> is a [[abelian|non-abelian]] group, in contrast to the other examples given here.

* The complement of a trefoil [[knot]] (overhand knot) in 3-dimensional space: <math>\pi_1(S^3 \setminus K)</math> is a [[abelian|non-abelian]] group, in contrast to the other examples given here.

These examples show that a wide range of groups are in fact the fundamental groups of some topological space.  In fact, every finitely presented group is the fundamental group of a topological space!  This observation makes possible the study of many aspects in group theory using techniques from topology.

These examples show that a wide range of groups are in fact the fundamental groups of some topological space.  In fact, every finitely presented group is the fundamental group of a topological space!  This observation makes possible the study of many aspects in group theory using techniques from topology.