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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)
##### Keywords:
classification of complete intersections
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