Hu, Po The cobordism of \(real\) manifolds. (English) Zbl 0939.55005 Fundam. Math. 161, No. 1-2, 119-136 (1999). The concept of Reality for vector bundles was introduced by M. F. Atiyah [Q. J. Math., Oxf. II. Ser. 17, 367-386 (1966; Zbl 0146.19101)]. A Real manifold then is a smooth manifold with a (smooth) \({\mathbb Z}/2\)-action and a Real structure on its stable normal bundle. The Real dimension of a Real manifold is an element in the representation ring \(R({\mathbb Z}/2)\cong {\mathbb Z}[\alpha]/(\alpha^2 = 1)\). Real cobordism was first considered by P. S. Landweber [Ann. Math., II. Ser. 86, 491-502 (1967; Zbl 0179.28503)], and about 10 years later M. Fujii [Math. J. Okayama Univ. 18, 171-188 (1976; Zbl 0334.55017)] introduced a Real Thom spectrum \(M{\mathbb R}\), which is a \({\mathbb Z}/2\)-equivariant spectrum in the sense of Lewis-May-Steinberger. However, one could not expect the Real cobordism groups to be isomorphic to the homotopy groups of \(M{\mathbb R}\), due to the lack of transversality. In the present paper the author constructs for any \(l\in {\mathbb N}\) an ordinary spectrum \(M{\mathbb R}_l\) such that the bordism group of \((k+l\alpha)\)-dimensional Real manifolds is isomorphic to the homotopy group \(\pi_kM{\mathbb R}_l\). The spectra \(M{\mathbb R}_l\) have a rather concrete definition, which also allows to perform some computations: they are the homotopy fibers of certain maps of spectra \(\delta_l\). The domain of \(\delta_l\) is a suspension of the suspension spectrum of the Thom space of the universal bundle over \(BO(l)\), and the range of \(\delta_l\) is a certain suspension of the homotopy orbit spectrum \(M{\mathbb R}_{h{\mathbb Z}/2}\). Reviewer: Michael Joachim (Münster) MSC: 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55P91 Equivariant homotopy theory in algebraic topology 57R90 Other types of cobordism 55P42 Stable homotopy theory, spectra Keywords:real cobordism; real manifolds Citations:Zbl 0146.19101; Zbl 0179.28503; Zbl 0334.55017 PDFBibTeX XMLCite \textit{P. Hu}, Fundam. Math. 161, No. 1--2, 119--136 (1999; Zbl 0939.55005) Full Text: EuDML