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Higher rank homogeneous Clifford structures

We give an upper bound for the rank r of homogeneous (even) Clifford structures on compact manifolds of non‐vanishing Euler characteristic. More precisely, we show that if r=2a·b with b odd, then r⩽9 for a=0, r⩽10 for a=1, r⩽12 for a=2 and r⩽16 for a⩾3. Moreover, we describe the four limiting cases and show that there is exactly one solution in each case.