§2 Topology and harmonic analysis.

§ 2.4. Some 2-manifolds, deck transformations.

Example 2: On the 2-sphere S^{2}, any two loops can be deformed into one another. The homotopy group \pi _{1}(S^{2}) is the trivial group consisting of the identity. The sphere S^{2} is simply connected. The torus T admits two different inequivalent loops. The first homotopy group becomes \pi _{1}(T)=C_{{\infty}}\times C_{{\infty}}. Its cover is the real plane R^{2}.

Since the simply connected manifolds of dimension n>2 can be classified as the Euclidean plane E^{n}, the sphere S^{n}, or the hyperbolic space H^{n}, we expect all the topological manifolds to appear as tilings of one of these objects.

Example 3: The torus T can be mapped to a square tiling of the Euclidean plane E^{2}. The unit square is shown in Fig.13 . A basis of the twofold Fourier analysis in coordinates (x,y) is given by

\displaystyle f_{{(k_{{n}},k_{m})}}(x,y)=\exp(i(k_{n}x+k_{m}y)), (13)
\displaystyle k_{n}=n\: 2\pi,n=0,\pm 1,\pm 2,...\: k_{m}=m\: 2\pi,m=0,\pm 1,\pm 2,...

After placing a topological manifold {\cal M} as a tiling on its universal cover, the matching rules of this tiling will reflect its connectivity and topology. The matching rules can be given as a set of deck operations which act on the universal cover and generate the tiling from an initial tile. The deck transformation group consists of these actions. We denote this group by {\rm deck}({\cal M}).

An important theorem in topology is:

Theorem[54] p. 196-8: The group deck({\cal M}) of deck transformations acting on the cover \tilde{\cal M} is isomorphic to the fundamental or first homotopy group,

deck({\cal M})\sim\pi _{1}({\cal M}). (14)

This theorem is not trivial since the structure of the two groups is very different.