Smooth structures on PL-manifolds of dimensions between 8 and 10

TL;DR

This paper classifies smooth structures on PL-manifolds of dimensions 8 to 10 using cohomology and Steenrod operations, computes homotopy inertia groups, and examines special cases like Lens spaces and real projective spaces.

Contribution

It provides a classification of smooth structures on PL-manifolds in dimensions 8 to 10 and computes associated homotopy inertia groups, extending previous understanding.

Findings

Classification of smooth structures on PL-manifolds of dimensions 8-10

Computation of homotopy inertia groups for these manifolds

Analysis of special cases such as Lens spaces and real projective spaces

Abstract

In this paper, we identify the concordance classes of smooth structures on $PL$-manifolds of dimension between $8$ and $10$ in terms of the cohomology and Steenrod operations. This leads to the computation of the homotopy inertia groups. Finally we discuss the special cases of Lens spaces and real projective spaces.