Principal bundles admitting almost invariant structures (Q941277)

Let \(\xi=(E,p,B,T^k)\) be a smooth principal bundle with projection \(p:E\to B\) and structure torus group \(T^k=(\mathbb R/\mathbb Z)^k\) over a manifold \(B\). Let a finite group \(\Delta\) act on \(B\) with right action \(R_\Delta:B\times\Delta\to B\). If \((V_1,h_1)\) and \((V_2,h_2)\) are two charts of \(\xi\), then \(\phi\colon V_1\cap V_2\to T^k\) is the transition function and \(\phi^*(\theta)\in\Omega^1(V_1\cap V_2;\mathbb R^k)\) is the transition form, where \(\theta\in\Omega^1(T^k;\mathbb R^k)\) is the canonical form. An atlas \({(V_i,h_i)}\) of \(\xi\) is called an almost \(\Delta\)-atlas if all \(V_i\) and all its transition forms are \(\Delta\)-invariant. An equivalence class \({\mathcal A}\) of almost \(\Delta\)-atlases is called an almost \(\Delta\)-structure on \(\xi\) and the pair \((\xi,{\mathcal A})\) is called an almost \(\Delta\)-bundle. In [Math. Notes 77, No.~4, 553--567 (2005); translation from Mat. Zametki 77, No. 4, 600--616 (2005; Zbl 1076.55005)], the authors constructed invariants of almost \(\Delta\)-bundles and calculated the group \({\mathcal B}(B, T^k, R_\Delta)\) in terms of the homology groups of the base \(B\). In this paper, they find conditions on the characteristic classes of a principal bundle \(\xi\) under which it possesses almost \(\Delta\)-structures. Also, they determine how many such bundles exist for fixed \(B\), \(T^k\), and \(R_\Delta\), and how many pairwise non-isomorphic almost \(\Delta\)-structures a given bundle \(\xi\) can admit.