Strong cohomological rigidity of a product of projective spaces
A toric manifold \(N\) is called cohomologically rigid if its integer cohomology determines its topological type among all toric manifolds. More precisely, if \(M\) is any other toric manifold, then the existence of a graded ring homomorphism \(\varphi:H^*(M) \to H^*(N)\) should imply the existence of a diffeomorphism (or homeomorphism) \(f:N\to M\). In a previous joint work with \textit{M. Masuda} [Trans. Am. Math. Soc. 362, No. 2, 1097--1112 (2010; Zbl 1195.57060)], the authors showed that products \(\prod_{i=1}^m {{\mathbb C}} P^{n_i}\) of complex projective spaces are cohomologically rigid (even in the more restrictive sense that there exists a diffeomorphism \(f\)).NEWLINENEWLINEIn this short paper the authors refine that statement; they show that for \(N=\prod_{i=1}^m {{\mathbb C}} P^{n_i}\) one can additionally require that the diffeomorphism \(f\) realizes \(\varphi\), in the sense that \(f^*=\varphi\); this property is called \textit{strong} cohomological rigidity. By the previously proven cohomological rigidity, the proof reduces to determining all graded automorphisms of \(H^*(\prod_{i=1}^m {{\mathbb C}}P^{n_i})\) and showing that each of them is realized by a self-diffeomorphism.NEWLINENEWLINEAs a corollary, it is shown that products of complex projective spaces are strongly cohomologically rigid in the class of quasitoric manifolds, in the sense that any isomorphism in cohomology is realized by a homeomorphism.
scientific article; zbMATH DE number 6066743