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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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