Higher spin Klein surfaces (Q2803008)

An \(m\)-spin Riemann surface is a pair \((P,e)\) in which \(P\) is a Riemann surface with a complex bundle line \(e: L \rightarrow P\) such that its \(m\)-th tensor power \(e^{\otimes m}: L^{\otimes m} \rightarrow P\) is isomorphic to the cotangent bundle of \(P\). The invariants of the \(m\)-spin surface \((P, e)\) are the genus \(g\) of \(P\), and the Arf invariant \(\delta \in \{0,1\}\).NEWLINENEWLINEThese concepts are extended to Klein surfaces, which are a generalization of Riemann surfaces allowing non-orientable and bordered surfaces. A Klein surface is a quotient \(P/\tau\) where \(P\) is a compact Riemann surface and \(\tau\) is an anti-holomorphic involution on \(P\). The category of Klein surfaces \((P, \tau)\) is isomorphic to that of real algebraic curves. The topological type of \((P, \tau)\) is a triple \((g, k, \epsilon)\), where \(g\) is the genus of \(P\), \(k\) is the number of boundary components of \(P/\tau\), and \(\epsilon\) is 0 or 1, according to the non-orientability or orientability of \(P/\tau\).NEWLINENEWLINEThen, an \(m\)-spin Klein surface is a Klein surface \((P, \tau)\) with an \(m\)-spin structure \((P, e)\) and an anti-holomorphic involution \(\beta : L \rightarrow L\) such that \(e\circ \beta = \tau \circ e\). Therefore, the \(m\)-spin Klein surface \((P, \tau, e, \beta)\) has the topological invariants \((g, k, \epsilon, m, \delta)\). NEWLINENEWLINENEWLINENEWLINE The aim of the paper under review is to prove that, for a given Klein surface \((P, \tau)\) of topological type \((g, k, \epsilon)\) with \(g \geq 2\), the number \(N(g, k, \epsilon, m, \delta)\) of \(m\)-spin Klein surfaces \((P, \tau, \epsilon, \beta)\) with the Arf invariant \(\delta\) depends only on the invariants \((g, k, \epsilon, m, \delta)\). Besides, that number \(N(g, k, \epsilon, m, \delta)\) is computed. This computation is made by counting the number of \(m\)-Arf functions. Given a Riemann surface \(P\), let \(\pi_1(P, p)\) be the fundamental group of \(P\) with respect to the point \(p\), and \(\pi_1^0(P)\) the set of non-trivial elements of \(\pi_1(P, p)\) that can be represented by simple closed curves. Then, an \(m\)-Arf function is a function \(\sigma : \pi_1^0(P) \rightarrow \mathbb{Z}/m\mathbb{Z}\) satisfying a number of conditions. In Section 3 the authors prove a correspondence between \(m\)-spin Klein surfaces and \(m\)-Arf functions, and in Section 4 they compute the \(m\)-Arf functions and hence the \(m\)-spin Klein surfaces.