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The Homotopy Types of Suspended Simply-connected $6$-manifolds

Pengcheng Li, Zhongjian Zhu

Abstract

Under the assumption that certain Adem cohomology operation acts trivially on $H^2(M;\mathbb{Z}/2)$, we determine the homotopy types of the triple suspension $Σ^3M$ of a simply-connected oriented closed topological(or smooth) $6$-manifold $M$, whose integral homology groups can have $2$-torsion.

The Homotopy Types of Suspended Simply-connected $6$-manifolds

Abstract

Under the assumption that certain Adem cohomology operation acts trivially on , we determine the homotopy types of the triple suspension of a simply-connected oriented closed topological(or smooth) -manifold , whose integral homology groups can have -torsion.
Paper Structure (6 sections, 18 theorems, 105 equations)

This paper contains 6 sections, 18 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Notations and some homotopy theory of $\mathbf{A}_{n}^{2}$-complexes
  3. Some lemmas on cohomology operations
  4. Homotopy decompositions of $\Sigma^3M$
  5. Reduction of coefficients
  6. Proofs of Theorem \ref{['thm:6mfds 2-local']} and \ref{['thm:6mfds total']}

Key Result

Theorem 1.1

Let $M$ be a simply-connected closed oriented $6$-manifold with $H_\ast(M)$ given by (HM). Assume that the Adem operation $\Psi$ acts trivially on $H^2(M;\mathbb{Z}/2^{})$. There are non-negative integers $k, t_i (i=0,1,2,3,4)$, $r_j (0\leq j\leq t_1)$, $\bar{r}_j (0\leq j\leq t_2)$, $\check{r}_j (0

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: cf. BH91
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 23 more