Use the results of Table 24.1 to show formally that for the Aharonov-Bohm effect the mapping discussed in Section 27.1 .4 from the electromagnetic gauge group manifold \(\mathrm{U}(1)\) to the plane pierced by the solenoid is non-trivial because it admits an infinity of topological sectors characterized by an integer winding number.

Data from Table 24.1

Data from Section 27.1 .4

Transcribed Image Text:

Table 24.1. Some properties of homotopy groups [2] (U(1)) =(S) = Z (U(1) U(1)) = (U(1)) (U(1)) = ZxZ (U(2)) = (S S) = (S)(S) = Z (SU(2)) =(S) = 1 Tin(Ax B) = In(A)XT,(B) n(S) (n> 1) T2(S) = (n2) T3(S) = 1 T(SO(n)) Z2 (n>2) n(S) 1 (n