Involutions in lens spaces
Let \(S^{2n+1}\subset\mathbb{C}^{n+1}\) be the unit sphere and consider a unitary map \(a:\mathbb{C}^{n+1}\to\mathbb{C}^{n+1}\) of period \(p\). The eigenvalues of \(a\) are then of the form \(\exp 2\pi\roman{i}q_k/p\) where \(0<q_k<p\). \(a\) defines a free cyclic action of \(\mathbb{Z}_p\) on \(S^{2n+1}\) and the orbit space is the lens space \( L_p(q):=L_p(q_0,\dots,q_n)\). The cohomology groups of \(L_p(q)\) (coefficients always being in \(\mathbb{Z}_2\)) is easily determined from the cell decomposition. Using the Smith-Gysin sequence the author determines the cohomology ring. If \(p\) is odd and if \(\alpha\) denotes the antipodal map the author estimates the cohomological dimension (with respect to Čech cohomology) of the antipodal coincidence set \(A(f):=\{y\in L_p(q)| \;f(y)=f(\alpha y)\}\) for a map \(f:L_p(q)\to\mathbb{R}^k\) by \(\dim A(f)=\dim\overline{A(f)}\geq 2n+1-k\) where \(\overline{A}\) refers to the quotient space by \(\alpha\). The author then generalizes this result to unitary periodic maps \(f:E\to E\) of a complex vector space bundle \(E\to B\) with a \(p\)-cyclic action which is free outside the zero section generated by \(f\) with \(p\) odd. Denote by \(S\to B\) the corresponding unit sphere bundle and \(L\to B\) the resulting lens space bundle with orbit space bundles \(\bar{S}\) and \(\bar{L}\). Call \(v_L\in H^1\bar{L}\) the characteristic class of the involution \(\alpha\) and \(u_S\) the characteristic class of the antipodal involution on \(S\). The author proves that \(v_L\mapsto u_S\) induces an isomorphism \(H^*\bar{L}\cong H^*\bar{S}\).
scientific article; zbMATH DE number 1317228