This paper constructs numerous examples of highly connected Poincaré complexes, each homotopy equivalent to a topological manifold yet not homotopy equivalent to any smooth manifold. Furthermore, we determine the homotopy type of any closed 2 k 2k -connected framed ( 4 k + 2 ) (4k+2) -manifold with Kervaire invariant one for k = 7 , 15 , 31 k=7,15,31 .
Wen Shen (Tue,) studied this question.