Torus actions on cohomology complex generalized Bott manifolds

Abstract

A torus manifold is a closed smooth manifold of dimension $2n$ having an effective smooth $T^n = (S^1)^n$-action with non-empty fixed points. Petrie \cite{petrie:1973} has shown that any homotopy equivalence between a complex projective space $\CP^n$ and a torus manifold homotopy equivalent to $\CP^n$ preserves their Pontrjagin classes. A \emph{generalized Bott manifold} is a closed smooth manifold obtained as the total space of an iterated complex projective space bundles over a point, where each fibration is a projectivization of the Whitney sum of a finite many complex line bundles. For instance, we obtain a product of complex projective spaces if all fibrations are trivial. If each fiber is $\CP^1$, then we call it an (ordinary) \emph{Bott manifold}. In this paper, we investigate the invariance of Pontrjagin classes for torus manifolds whose cohomology ring is isomorphic to that of generalized Bott manifolds. We show that any cohomology ring isomorphism between two torus manifolds whose cohomology ring is isomorphic to that of a product of projective spaces preserves their Pontrjagin classes, which generalizes the Petrie's theorem. In addition, we show that any cohomology ring isomorphism between two torus cohomology Bott manifolds preserves their Pontrjagin classes. As a corollary, there are at most a finite number of torus manifolds homotopy equivalent to either a given product of complex projective space or a given Bott manifold.