The authors characterize the simple homotopy manifold set via algebraic K-theory and surgery obstructions, then construct infinite families of even-dimensional manifolds homotopy equivalent to S^1 times a lens space that are pairwise not simple homotopy equivalent.
[IN12] Humio Ichimura and Shoichi Nakajima,On the 2-part of the class numbers of cyclotomic fields of prime power conductors, J
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Simple homotopy types of even dimensional manifolds
The authors characterize the simple homotopy manifold set via algebraic K-theory and surgery obstructions, then construct infinite families of even-dimensional manifolds homotopy equivalent to S^1 times a lens space that are pairwise not simple homotopy equivalent.