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Bordism theory with reality and duality theorem of Poincaré type. (English) Zbl 0702.57016

The author studies the cobordism theory of closed differentiable manifolds with a given real structure on the stable tangent bundle. A \(\tau\)-manifold (M,t) is a \(C^{\infty}\)-manifold M together with a \(C^{\infty}\)-involution t: \(M\to M\), and a \(\tau\) map is an equivariant map between \(\tau\)-manifolds. A real structure on a \(\tau\)-vector bundle over a \(\tau\)-manifold (i.e. projection is a \(\tau\)-map) is a bundle isomorphism J with \(J^ 2=-id,\quad \tau J=(-J)\tau\) and \(proj(- J)=proj.\) The author proves a Poincaré duality theorem between the bordism and cobordism theory of manifolds with real structures.
Reviewer: Y.F.Wong

MSC:

57R90 Other types of cobordism
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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