Some Elementary Gauge Theory Concepts by Chan Hong-Mo, Tsou Sheung Tsun

By Chan Hong-Mo, Tsou Sheung Tsun

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For topics dealt with in this book, we shall be interested mainly in the low order homotopy groups of spheres Sn and of classical and exceptional groups. We defer the discussions of the homotopy groups of the classical and exceptional Lie groups till the next section. 2. 4), we can deduce: 7r,-(S2) = Xi-xiS^+ViiS3) 7Ti(54) = 7T i 7Ti(58) = ir i l (5 7 ) + jr i (5 15 ) 1 (5 3 if a > 2 )+X i (5 7 ) if >2 if * > 2, and the first of these relations give the following more succinct result: 7r((53) = 7r (52) if * > 3.

For example, the two components of 0(3) can be identified with the two elements of the group 2, the component where the determinant equals 1 corresponding to 0 in 2 and the component where the determinant equals —1 corresponding to 1 in Furthermore, for n > 2, irn{X) is always abelian. 7Ti(^0 is in general not abelian. For example, if X is a Riemann surface with genus > 1, then ^(A") is not abelian. However, the fundamental group of a Lie group is always abelian. For these results and many of those that follow, the reader will find a particularly interesting chapter in Steenrod (1951).

Secondly, we wish to collect together for easy reference some related mathematical results in anticipation of possible future applications within the context of our subject. This material is scattered over a wide area in the mathematical literature and may be tedious to locate for those unfamiliar with it. For this reason, we think it is helpful to have at hand a concise summary of possibly useful known results, although it is obviously not easy to know exactly what is likely to be needed. This summary will be given mostly in the form of tables.

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