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A variant of \(K\)-theory and topological T-duality for real circle bundles. (English) Zbl 1320.19001
Let \(X\) be a topological space, \(\pi:E\to X\) a principal circle bundle, and \(h\in H^3(E,\mathbb{Z})\) a cohomology class. For each pair \((E,h)\) as above, there exist another pair \((E',h')\) such that the Chern classes of \(E,E'\) satisfy \(c_1(E)=\pi'_*h', c_1(E')=\pi_*h,\) and moreover the twisted \(K\)-theory groups satisfy \(K^{h+n}(E)\simeq K^{h'+n-1}(E')\). The pair \((E',h')\) is unique up to isomorphism. This is a version of “topological T-duality”, cf., e.g. [U. Bunke and T. Schick, Rev. Math. Phys. 17, No. 1, 77–112 (2005; Zbl 1148.55009)].
The present paper extends the topological T-duality to the category of spaces endowed with a \(\mathbb{Z}/2\mathbb{Z}\)-action, Real circle bundles, \(\mathbb{Z}/2\mathbb{Z}\)-twisted cohomology, and twisted equivariant \(K\)-theory. (a Real circle bundle is a principal \(\mathbb{S}^1\)-bundle over a \(\mathbb{Z}/2\mathbb{Z}\)-space with involution \(\tau\), together with a lift of \(\tau\) to an involution \(\tau\) of the total space \(E\) such that \(\tau(\xi u)=\tau(\xi)u^{-1}\) for every \(\xi\in E\) and \(u\in \mathbb{S}^1\)).
Previous results were obtained by D. Baraglia [Pure Appl. Math. Q. 10, No. 3, 367–438 (2014; Zbl 1318.19009)] for free actions. The T-duality statement from this paper, extending Baraglia’s result, was independently proved with different methods by V. Mathai and J. Rosenberg [“T-duality for circle bundles via noncommutative geometry”, Adv. Theor. Math. Phys. (to appear), arXiv:1306.4198].

MSC:
19L50 Twisted \(K\)-theory; differential \(K\)-theory
55N91 Equivariant homology and cohomology in algebraic topology
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