# Journal of the Ramanujan Mathematical Society

Volume 28A, Issue SPL, July – Special Issue 2013 pp. 55–74.

Formality of certain CW complexes and applications to Schubert varieties and torus manifolds**Authors**: Prateep Chakraborty and Parameswaran Sankaran

**Author institution:**The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India

**Summary:**Let X be a path connected topological space having the homotopy type of a CW complex. We show that X is formal if H* (X;Q) is generated by Hk (X; Q) for some even integer k ≠ 2 where dim Hk (X; Q) < ∞. As an application we show that any union of Schubert varieties in a generalized complex flag variety G/B, where G is a complex semisimple Lie group and B, a Borel subgroup is formal. The same result is also obtained for Schubert ‘varieties’ in quaternionic complete flag manifolds. Also we obtain a new proof of a result of Panov and Ray that any torus manifold over a homology polytope where the torus action is locally standard is formal.

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