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Cohomology algebra of orbit spaces of free involutions on lens spaces. (English) Zbl 1292.57030

Let \(p\geq 2\) be a positive integer and \(q_1,q_2,\dots, q_m\) be integers prime to \(p\). Let \(L^{2m-1}_p(q_1,q_2,\dots, q_m)\) denote the orbit space of the action of the cyclic group \(G= \mathbb{Z}_2\) on \(S^{2m-1}\subset \mathbb{C}^m\) which sends \((z_1,z_2,\dots, z_m)\) to \((e^{2\pi iq_1/p} z_1,\dots, e^{2\pi iq_m/p} z_m)\).
The main theorem on this paper completely determines the possible cohomology algebras \(H^*(X/G;\mathbb{Z}_2)\) arising from free \(G\) actions on a finitistic space \(X\) with the \(\mathbb{Z}_2\) cohomology of \(L^{2m-1}_p(q_1,q_2,\dots, q_m)\). There are five possible truncated-polynomial algebras with 1, 2, 3, or 4 generators. For example, if \(p\) is odd then \(H^*(X/G;\mathbb{Z}_2)= \mathbb{Z}_2[x]/(x^{2m})\), \(\deg(x)= 1\), and if \(p\) is even, \(p\not\equiv 0\pmod 4\), then \(H^*(X/G; \mathbb{Z}_2)= \mathbb{Z}[x,y]/(x^A2, y^m)\), \(\deg(x)= 1\) and \(\deg(y)= 2\). If \(p\equiv 0\pmod 4\), the description of the cohomology algebra depends on differentials in the Leray spectral sequence of the Borel fibration \(X\to X_G\to B_G\). The paper includes a Borsuk-Ulam type application of the main theorem. If \(m\geq 3\) and \(X\) is a finitistic space with the \(\mathbb{Z}_2\) cohomology of \(L^{2m-1}_p(q_1,q_2,\dots, q_m)\) and \(X\) admits a free involution, then there does not exist a \(\mathbb{Z}_2\) equivariant map \(S^n\to X\) if \(n\geq 2m\).

MSC:

57S17 Finite transformation groups
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
55M20 Fixed points and coincidences in algebraic topology
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