  The study of higher order interactions in the dynamics of Kuramoto
oscillators has been a topic of intense recent research. Arguments based on
dimensional reduction using the Ott-Antonsen ansatz show that such interactions
usually facilitate synchronization, giving rise to bi-stability and hysteresis.
Here we show that three body interactions shift the critical coupling for
synchronization towards higher values in all dimensions, except $D=2$, where a
cancellation occurs. After the transition, three and four body interactions
combine to facilitate synchronization. Similar to the 2-dimensional case,
bi-stability and hysteresis develop for large enough higher order interactions.
We show simulations in $D=3$ and $4$ to illustrate the dynamics.
