  We consider consequences of triviality of cobordism classes and anomaly
cancellation in supergravity theories in $d>6$. We argue that this leads to the
existence of certain defects which we call "I-folds" (a generalization of
orientifolds). The requirement that compactifications to lower dimensions
involving these defects be anomaly free leads to conditions on the higher
dimensional theory. We show that in theories with 16 supercharges in $d>6$ this
leads to restrictions on the rank of the allowed gauge groups and thus provides
an explanation for the observed restrictions in known string theory
constructions. In particular, in eight and nine dimensions the only solutions
to our constraints are precisely the ones realized in string theory
compactifications. We also use these techniques to place constraints on the
global structure of the gauge group in eight and nine dimensions.
