Generators of the unoriented bordism ring which are fibered over products of projective spaces \(RP(2^ r-1)\)
type: Item
instance of: scholarly article
title: Generators of the unoriented bordism ring which are fibered over products of projective spaces \(RP(2^ r-1)\)
zbMATH Open document ID: 0887.57038
DOI: 10.1007/BF02567953
author: Oswald Gschnitzer
published in: Manuscripta Mathematica
publication date: 1997-04-08T00:00:00Z
full work available at URL: https://eudml.org/doc/156228
review text: The author constructs simple generators of the unoriented cobordism ring \(M O_*\). Fix an integer \(0<k\). Suppose \(n\) is a positive integer such that \(n+1\) is not a power of 2. Write \(n=a 2^k+m\) where \(0\leq a\), \(0\leq m<2^k\). The author defines an integer \(g_k(n)\) \((g_k (n) \leq n-a)\). (When \(k=1\), \(a= [n/2]\), \(g_1(n)\) is either \(a\) or \(a-1)\). The author proves there is an \(n\)-dimensional manifold \(X^n\) representing an indecomposable class in \(MO_n\) and \(X^n\) is fibered over a \(g_k(n)\)-fold product of projective spaces, \(\Pi_i {\mathbf R} P(2^{r_i}-1)\), where \(1\leq r_i\leq k\). When \(k=1\), all \(r_i=1\), and all \({\mathbf R} P(2^{r_i} -1)\) are circles. In this case, the author's result gives representing manifolds fibered over \(g_1(n)\)-dimensional tori. This enhances a quarter-century old result of Stong.
Mathematics Subject Classification ID: 57R75
Mathematics Subject Classification ID: 55N22
zbMATH DE Number: 980693
zbMATH Keywords: unoriented bordism
zbMATH Keywords: generators
zbMATH Keywords: projective spaces
zbMATH Keywords: fibred
reviewed by: David C. Johnson
MaRDI profile type: Publication
cites work: Manifolds and Stiefel-Whitney Classes
cites work: Diffeomorphisms of period two
cites work: Differentiable periodic maps. 2nd ed
cites work: On Fibering of Cobordism Classes
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description: scientific article; zbMATH DE number 980693