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Surgery and duality. (English) Zbl 0935.57039
This paper describes a new context in which to do surgery on manifolds. As the author describes it in rough terms, one compares \(n\)-dimensional manifolds with a topological space having a chosen \(k\)-skeleton for some \(k\geq [n/2]\). A bit more precisely, one begins with a map \(B\to BO\) and does surgery on maps \(M \to B\) classifying the normal bundle with \(M\) homotopy equivalent to \(B\) up to dimension \(k\).
The initial example is the classification of complete intersections of complex dimension \(n\) for which \(B\) looks like \(CP^\infty\) up to dimension \(n\). Other specific applications are given for 4-dimensional topological manifolds which are Spin and have \(\pi_1= Z\) and for simply connected 7-dimensional homogeneous spaces.
A sample of a good general result is that two closed \(2q\)-dimensional manifolds with the same Euler characteristic and same normal \((q-1)\)-type admitting bordant normal \((q-1)\)-smoothings are diffeomorphic after taking the connected sum with some copies of \(S^q\times S^q\).

MSC:
57R65 Surgery and handlebodies
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
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