Comparing the homotopy types of the components of Map(S4,BSU(2))

We classify the homotopy types of the connected components of Map(S4,BSU(2)). Since image, the components are Mapk(S4,BSU(2)) consisting of maps of degree image. Clearly, Mapk(S4,BSU(2)) is homotopy equivalent to Map−k(S4,BSU(2)), by composition with the antipodal map. We prove the converse, i.e. Mapk(S4,BSU(2))similar, equalsMapl(S4,BSU(2))impliesk=±l. The result is of interest in gauge theory because the Mapk(S4,BSU(2)) are the classifying spaces of the gauge groups of SU(2) bundles over S4.