Strong cohomological rigidity of a product of projective spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1247.57025
DOI: 10.4134/BKMS.2012.49.4.761
author: Suyoung Choi
author: Dong Youp Suh
published in: Bulletin of the Korean Mathematical Society
publication date: 2012-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.4791
full work available at URL: http://www.mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 6066743
zbMATH Keywords: product of projective spaces
zbMATH Keywords: generalized Bott manifold
zbMATH Keywords: strong cohomological rigidity
zbMATH Keywords: toric manifold
zbMATH Keywords: quasitoric manifold
MaRDI profile type: Publication
arXiv ID: 0912.4791
OpenAlex ID: W2964079477
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title: Strong cohomological rigidity of a product of projective spaces
review text: 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.
reviewed by: Oliver Goertsches
description: scientific article; zbMATH DE number 6066743