[Paper Review] Torus actions on cohomology complex generalized Bott manifolds

Suyoung Choi|arXiv (Cornell University)|Jan 29, 2011

Advanced Combinatorial Mathematics11 references3 citations

TL;DR

This paper establishes that Pontrjagin classes are preserved under cohomology ring isomorphisms for torus manifolds with cohomology isomorphic to generalized Bott manifolds or products of complex projective spaces. By extending Petrie's theorem to this broader class of manifolds, it proves that such isomorphisms preserve Pontrjagin classes, implying finitely many torus manifolds exist up to homotopy equivalence in these cases.

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.

Motivation & Objective

• To generalize Petrie's theorem on Pontrjagin class invariance from complex projective spaces to torus manifolds with cohomology isomorphic to generalized Bott manifolds.
• To investigate whether cohomology ring isomorphisms between torus manifolds preserve their Pontrjagin classes when the cohomology ring matches that of a product of complex projective spaces.
• To determine whether such isomorphisms also preserve Pontrjagin classes in the special case of torus cohomology Bott manifolds.
• To establish finiteness results for torus manifolds homotopy equivalent to a given product of complex projective spaces or a given Bott manifold.

Proposed method

• The authors analyze torus manifolds with cohomology rings isomorphic to those of generalized Bott manifolds, using the structure of iterated projectivizations of Whitney sums of complex line bundles.
• They apply techniques from equivariant cohomology and torus actions to study the fixed point data and characteristic classes of such manifolds.
• The proof relies on the fact that the cohomology ring determines the topological type in the case of generalized Bott manifolds, particularly through the structure of the associated Bott tower.
• The authors use the invariance of Pontrjagin classes under homotopy equivalences and extend this via cohomological rigidity to the class of torus manifolds with generalized Bott manifold cohomology.
• They leverage the fact that the cohomology ring of a generalized Bott manifold is generated by first Chern classes of line bundles, enabling explicit computation of characteristic classes.
• The argument proceeds by showing that any cohomology ring isomorphism between such manifolds must preserve the integral cohomology classes that represent the Pontrjagin classes.

Experimental results

Research questions

• RQ1 Does the invariance of Pontrjagin classes under cohomology ring isomorphisms hold for torus manifolds whose cohomology ring is isomorphic to that of a product of complex projective spaces?
• RQ2 Can Petrie's theorem on Pontrjagin class preservation be extended to generalized Bott manifolds via cohomology ring isomorphisms?
• RQ3 Is the Pontrjagin class preserved under cohomology ring isomorphisms between torus cohomology Bott manifolds?
• RQ4 Are there only finitely many torus manifolds homotopy equivalent to a given product of complex projective spaces or a given Bott manifold?

Key findings

• Any cohomology ring isomorphism between two torus manifolds with cohomology isomorphic to a product of complex projective spaces preserves their Pontrjagin classes.
• The same preservation property holds for torus cohomology Bott manifolds, extending Petrie's result beyond complex projective spaces.
• The Pontrjagin classes are invariant under cohomology ring isomorphisms in the broader class of torus manifolds with generalized Bott manifold cohomology.
• The finiteness of torus manifolds homotopy equivalent to a given product of complex projective spaces or a given Bott manifold follows from the cohomological rigidity of these classes.
• The results demonstrate that the cohomology ring structure of these manifolds is sufficient to determine the Pontrjagin classes up to isomorphism.
• The study confirms that the topological invariants encoded in the cohomology ring are strong enough to control characteristic classes in this class of torus manifolds.