Semicharacteristics of oriented Grassmannians
type: Item
instance of: scholarly article
title: Semicharacteristics of oriented Grassmannians
zbMATH Open document ID: 0573.57007
DOI: 10.1016/0022-4049(84)90029-X
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let M be a closed manifold of dimension \(2m+1\). The semicharacteristic of M with respect to a field F is defined to be \[ S\chi_ F(M)=\sum^{m}_{0}(-1)^ i\quad \dim_ F H^ i(M;F) mod 2. \] If F is of characteristic not 2, then one assumes that M is oriented. In this paper the author proves that the semicharacteristics of \(G=G^+_ k({\mathbb{R}}^{n+k})\), the oriented Grassmannians, for nk odd are independent of the field and given by \(\left( \begin{matrix} [n]+[k]\\ [k]\end{matrix} \right)\) (mod 2). For nk\(\equiv 1\) mod 4 the result is deduced from a theorem of \textit{G. Lusztig, J. Milnor} and \textit{F. P. Peterson} [Topology 8, 357-359 (1969; Zbl 0165.263)], and by examing the cohomology ring of G. For the case nk\(\equiv 3\) mod 4 the result is proved by constructing a cobordism (V,T) between a copy of G and a projective space bundle whose base is oriented Grassmannians of dimension \(\equiv 1\) mod 4 and by applying his formula \(S\chi (\partial V)=\chi (F)+\chi (F\cap F)\), where F is the fixed point set of an involution T and \(F\cap F\) is the self-intersection of F in V.
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R85
Mathematics Subject Classification ID: 57T15
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 3915244
zbMATH Keywords: closed manifold
zbMATH Keywords: semicharacteristic
zbMATH Keywords: oriented Grassmannians
zbMATH Keywords: projective space bundle
author: R. E. Stong
MaRDI profile type: Publication
cites work: Q3961493
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cites work: Semi-characteristics and cobordism
cites work: On stable parallelizability of \(G_{k,n}\) and related manifolds
cites work: Q4046794
cites work: Cup products in Grassmannians
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description: scientific article; zbMATH DE number 3915244