Manifold structures on highly connected Poincar\'e complexes

TL;DR

This paper constructs examples of highly connected Poincaré complexes that are topological manifolds but not smooth, and determines the homotopy types of certain framed manifolds with Kervaire invariant one for specific dimensions.

Contribution

It introduces new examples of highly connected Poincaré complexes not homotopy equivalent to smooth manifolds and classifies homotopy types of specific framed manifolds with Kervaire invariant one.

Findings

Constructed numerous examples of such complexes.

Determined homotopy types for manifolds with Kervaire invariant one in specific dimensions.

Showed differences between topological and smooth manifold structures.

Abstract

This paper constructs numerous examples of highly connected Poincar\'{e} 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 $2k$-connected framed $(4k+2)$-manifold with Kervaire invariant one for $k=7,15,31$.