On the level of projective spaces
type: Item
instance of: scholarly article
title: On the level of projective spaces
zbMATH Open document ID: 0634.10020
DOI: 10.1007/BF02564448
author: Stephan Stolz
author: Albrecht Pfister
published in: Commentarii Mathematici Helvetici
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140086
review text: By definition [see \textit{Z. D. Dai} and \textit{T. Y. Lam}, ibid. 59, 376--424 (1984; Zbl 0546.10017)] the level of a topological space \(X\) with a fixed point free involution \(i\) is the number \(s(X,i)=\min \{n:\) there is a \(\mathbb Z/2\)-equivariant map \(f: X\to S^{n-1}\}\). Here \(S^{n-1}\) has a \(\mathbb Z/2\)-action given by the antipodal involution. The Borsuk-Ulam theorem implies \(s(S^{n-1},-)=n\). Results on the levels of Stiefel manifolds are given in (Dai and Lam, loc. cit.). If \(n\) is odd, the real projective space \(\mathbb{RP}^n\) has a fixed point free involution. Explicitly, multiplication by the complex number \(i\) gives a \(\mathbb Z/4\)-action on \(S^{2m-1}\subset \mathbb C^m\), which induces a \(\mathbb Z/2\)-action on the quotient space \(\mathbb{RP}^{2m-1} = S^{2m-1}/\mathbb Z/2\). Theorem. \[ m+1\leq s(\mathbb{RP}^{2m-1},i)\leq (3m+1)/2. \] The lower bound is proved using explicit knowledge of the \(K\)-theory of lens spaces. The upper bound is found by constructing an equivariant map built from the imaginary parts of an anisotropic system of complex quadratic forms. By modifying the proofs the authors provide similar estimates for the levels of the complex projective spaces.
reviewed by: Daniel B. Shapiro
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55M35
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 4031787
zbMATH Keywords: level
zbMATH Keywords: real projective space
zbMATH Keywords: fixed point free involution
zbMATH Keywords: complex projective spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4031787