On the geometry of moduli spaces of anti-self-dual connections
Let \(X\) be a smooth, simply connected, projective complex surface. Let \(M_k^f(X)\) be the moduli space of rank \(2\) framed instantons with second Chern class \(k\) and let \(C_k^f(X)\) be the space of all corresponding framed connections modulo gauge equivalence.NEWLINENEWLINEA generalization of the Atiyah-Jones conjecture says that the inclusion \(M_k^f(X)\to C_k^f(X)\) induces isomorphisms on homology and homotopy for sufficiently large \(k\). In [Adv. Math. 218, No. 4, 1027--1050 (2008; Zbl 1191.14016)], \textit{E. Gasparim} showed that if this holds for homology on some surface \(X\) then it also holds for homology on a blow up of \(X\).NEWLINENEWLINEIn the paper under review the authors show that if \(\pi_1(M_k^f(X))\) is polycyclic-by-finite and the Atiyah-Jones conjecture holds for \(\pi_1\) on some surface \(X\) then it also holds for \(\pi_1\) on a blow up of \(X\).
scientific article; zbMATH DE number 6023678
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