A real surface is a pair \((X,\sigma)\) consisting of a compact connected Riemann surface \(X\) equipped with an antiholomorphic involution \(\sigma\). A classification of real surfaces was studied by G. Weichold in his Dissertation submitted to Universität Leipzig in 1883. Weichold proved that the isomorphism type of a real surface \((X,\sigma)\) is completely determined by the triple of numbers \((g(X), r(X), a(X))\), where \(g(X)\) is the genus of \(X\), \(r(X)\) is the number of path components of the fixed set \(X^{\sigma}\) for the involution \(\sigma\) on \(X\), and \(a(X)=0\) if the orbit set \(X/\sigma\) is orientable, otherwise \(a(X)=1\).
The topology of gauge groups over real surfaces is of great interest and has been studied by many authors, among others by I. Biswas, J. Huisman and J. Hurtubise [Math. Ann. 347, No. 1, 201–233 (2010; Zbl 1195.14048)]. These authors classified the real vector bundles over real surfaces and calculated some of the low-dimensional homotopy groups of the associated gauge groups.
In the present paper, the author extends the calculations of homotopy groups of gauge groups by Biswas, Huisman and Hubertubise by providing homotopy decompositions of the gauge groups into products of known factors. The paper is carefully and well written.