The topology of is the product of the topology of SO(n) and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of -dimensional Euclidean space. Thus, the group has the same fundamental group as SO(n), that is, Z for and Z2 for . Sections 13.2 and 13.3 In particular this means that , unlike , is not simply connected, for n greater than 1.
the toPOlogy Of iS THE pRoDuct of The toPology oF so(n) anD The ToPoLogY Of tHe Group oF symmetric MAtRIceS with pOSitiVE eIgeNValueS aNd unit deTeRMiNant. SiNCe The latter matricES can be UniquELy eXprEssEd As The eXPOnentiaL oF symmeTric traCELEsS MatRIces, Then this latter topolOgy is That OF -dImEnSIOnal euClidEaN spAcE. tHuS, tHe Group HaS the Same funDaMeNtAl GrouP As so(N), that iS, Z fOr aNd Z2 for . sEctIons 13.2 And 13.3 iN ParTicuLaR ThiS meANs tHat , UnLikE , is nOt SImply cOnnectED, for n greater than 1.