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$C$-triviality of manifolds of low dimensions

Shubham Sharma, Animesh Renanse

Abstract

A space $X$ is said to be $C$-trivial if the total Chern class $c(α)$ equals $1$ for every complex vector bundle $α$ over $X$. In this note we give a complete homological classification of $C$-trivial closed smooth manifolds of dimension $< 7$. In dimension $7$ we give a complete classification of orientable $C$-trivial manifolds and in the non-orientable case we give necessary homological conditions for the manifold to be $C$-trivial. Our main tool is the Atiyah-Hirzebruch spectral sequence and orders of its differentials.

$C$-triviality of manifolds of low dimensions

Abstract

A space is said to be -trivial if the total Chern class equals for every complex vector bundle over . In this note we give a complete homological classification of -trivial closed smooth manifolds of dimension . In dimension we give a complete classification of orientable -trivial manifolds and in the non-orientable case we give necessary homological conditions for the manifold to be -trivial. Our main tool is the Atiyah-Hirzebruch spectral sequence and orders of its differentials.

Paper Structure

This paper contains 4 sections, 17 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Proof of general results
  3. Calculations for low dimensional manifolds
  4. Applications of the results

Key Result

Theorem 1.1

atiyah For a finite $CW$-complex $X$, the $9$-fold suspension $\Sigma^9X$ is $W$-trivial. ∎

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 28 more