Positive intermediate curvatures and Ricci flow.

We show that, for any n\geq 2, there exists a homogeneous space of dimension d=8n-4 with metrics of Ric_{\frac {d}{2}-5}>0 if n\neq 3 and Ric_6>0 if n=3 which evolve under the Ricci flow to metrics whose Ricci tensor is not (d-4)-positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of Böhm and Wilking [Geom. Funct. Anal. 17 (2007), pp. 665–681] in the case of n=2. [ABSTRACT FROM AUTHOR]