Algebraic Topology, Direct Product of Fundamental Groups

Direct Product of Fundamental Groups

The direct product of spaces has, as its fundamental group, the direct product of the component groups.

A loop is a continuous function on S¹, that maps a designated point on the circle to the base point in space.

A homotopy between loops is a continuous function on S¹ cross I that "connects" two such loops.

Let P be the product of component spaces S_i. Fix a base point c in P, having projections c_i in S_i. Remember that a function into P is continuous iff it is continuous into each S_i, thus a loop in P, based at c, defines, and is defined by, a combination of loops from the component spaces S_i, based at c_i. Furthermore, a homotopy between loops in P defines, and is defined by, homotopies in the component spaces, between the component loops. As a set, the loop classes of P are the direct product of the loop classes of each S_i.

Concatenate two loops in P, and project down to S_i, and we are really concatenating the two component loops in S_i. The group operator is applied per component, and π₁(P) is the direct product of π₁(S_i).

Fundamental Group of the Annulus

topologically, the annulus is homeomorphic to the circle cross an interval. Its fundamental group is Z cross 1, or Z. The projection of a loop onto the circle determines the homotopy class. The projection onto the interval is of no consequence.

The plane minus the origin is a special case of the annulus, and has a fundamental group of Z. In an earlier section we called this the winding number of the curve.

Fundamental Group of the Torus

Cross the circle with itself n times to get the n dimensional torus, denoted Tⁿ. This space has the fundamental group Zⁿ, or n copies of Z running in parallel.

As a special case, consider T², the traditional torus, which has the shape of a doughnut. The fundamental group is Z². Given a closed loop on the torus, the first integer in Z² measures the number of times the loop traverses the length of the torus, while the second integer measures the number of windings around the tube.

For each dimension n, Tⁿ is a different space, having a different homotopy group. And these are all different from real space, which has the trivial homotopy group.