On the level of projective spaces
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.
scientific article; zbMATH DE number 4031787