Cohomological rigidity of real Bott manifolds

Yoshinobu Kamishima, Mikiya Masuda
2009 Algebraic and Geometric Topology  
A real Bott manifold is the total space of an iterated RP 1 -bundle over a point, where each RP 1 -bundle is the projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z=2-coefficients are isomorphic. A real Bott manifold is a real toric manifold and admits a flat Riemannian metric invariant under the natural action of an elementary abelian 2-group. We also prove that the converse is true, namely a real
more » ... rue, namely a real toric manifold which admits a flat Riemannian metric invariant under the action of an elementary abelian 2-group is a real Bott manifold. positive answer, but no counterexample is known and there are some partial affirmative solutions to the problem (see [6; 16; 17]). The set X.R/ of real points in a toric manifold X is called a real toric manifold. It appears as the fixed point set of complex conjugation on X . For example, when X is a complex projective space CP n , the subspace X.R/ is a real projective space RP n . It is known that for any toric manifold X where Z denotes the integers and Z=2 D f0; 1g, and one may ask the same question as the rigidity problem above for real toric manifolds with Z=2-coefficients, namely: Cohomological rigidity problem for real toric manifolds Are two real toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings with Z=2-coefficients are isomorphic as graded rings? In this paper we are concerned with a sequence of RP 1 -bundles such that M i ! M i 1 for i D 1; : : : ; n is the projective bundle of a Whitney sum of two real line bundles over M i 1 , where one of the two line bundles may be assumed to be trivial without loss of generality because projectivizations P .E/ and P .E˝L/ are diffeomorphic for any real vector bundle E and line bundle L over a smooth manifold. Grossberg and Karshon [9] considered the sequence above in the complex case and named it a Bott tower of height n. Following them, we call the sequence above a real Bott tower of height n. Each M i in the tower (1-1) is a real toric manifold. We call M i a real Bott manifold. There are many choices of line bundles in the tower (1-1) so that real Bott towers produce many real Bott manifolds. We note that even if real Bott towers of height n are different, their top manifolds of dimension n might be diffeomorphic. The main purpose of this paper is to prove the following which answers the cohomological rigidity problem affirmatively for real Bott manifolds. Theorem 1.1 Two real Bott manifolds are diffeomorphic if their cohomology rings with Z=2-coefficients are isomorphic as graded rings. Although real toric manifolds have similar properties to toric manifolds, there is one major difference, which is that a real toric manifold is not simply connected while a
doi:10.2140/agt.2009.9.2479 fatcat:yf3657xanre25cplqhgq6lec2q