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

Samik Basu, Ramesh Kasilingam, Priyanka Magar-Sawant

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.

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

Abstract

In this paper, we identify the concordance classes of smooth structures on -manifolds of dimension between and 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.
Paper Structure (9 sections, 21 theorems, 35 equations)

This paper contains 9 sections, 21 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Organization
  4. Smooth structures on $8,9$-manifolds
  5. Smooth structures on $10$-manifolds
  6. Inertia group of $M^n$ for $8\leq n\leq 10$
  7. Concordance inertia group
  8. Homotopy inertia group
  9. Inertia group of $\mathbb{R}\mathrm{P}^{10}$

Key Result

Theorem A

The smooth concordance structure set $\mathcal{C}(M)$ for manifolds $M$ with $8 \leq \dim(M) \leq 10$ is explicitly determined in terms of the action of Steenrod operations on the cohomology of $M$.

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • ...and 30 more