Surgery and duality

Matthias Kreck

Review from Zentralblatt MATH:

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 $S^q\times S^q$.

Reviewed by R.E.Stong

Keywords: classification of complete intersections

Classification (MSC2000): 57R65 57N13 57N15

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