Comparing the homotopy types of the components of Map(S-4, BSU(2))

We classify the homotopy types of the connected components of Map(S-4,BSU(2 )). Since (pi4)(BSU(2)) = Z, the components are Map(K)(S-4,BSU(2)) consisti ng of maps of degree k is an element of Z. Clearly, Map(k)(S-4,BSU(2)) is h omotopy equivalent to Map(-k)(S-4,BSU(2)), by composition with the antipoda l map. We prove the converse, i.e. Map(k)(S-4, BSU(2)) similar or equal to Map(l)(S-4,BSU(2)) double right arrow k=+/-l. The result is of interest in gauge theory because the Map(k)(S-4, BSU(2)) are the classifying spaces of the gauge groups of SU(2) bundles over S-4. (C) 2001 Elsevier Science B.V. All rights reserved.