Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The homotopy decomposition of the suspension of a non-simply-connected $5$-manifold

Pengcheng Li, Zhongjian Zhu

Abstract

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $5$-manifolds. As applications, we compute the reduced $K$-groups of $M$ and show that the suspension map between the third cohomotopy set $π^3(M)$ and the fourth cohomotopy set $π^4(ΣM)$ is a bijection.

The homotopy decomposition of the suspension of a non-simply-connected $5$-manifold

Abstract

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth -manifolds. As applications, we compute the reduced -groups of and show that the suspension map between the third cohomotopy set and the fourth cohomotopy set is a bijection.
Paper Structure (9 sections, 31 theorems, 108 equations)

This paper contains 9 sections, 31 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some homotopy theory of $\mathbf{A}_n^2$-complexes
  4. Basic analysis methods
  5. Homology Decomposition of $\Sigma M$
  6. Proof of Theorem \ref{['thm:main']} and \ref{['thm:dbsusp']}
  7. Some Applications
  8. Reduced $K$-groups
  9. Cohomotopy sets

Key Result

Theorem 1.1

Let $M$ be an orientable smooth closed $5$-manifold with $H_\ast(M)$ given by (HM). Let $T_2\cong \bigoplus_{j=1}^{t_2}\mathbb{Z}/2^{r_j}$ be the $2$-primary component of $T$ and suppose that $H$ contains no $2$- or $3$-torsion. There exist integers $c_1,c_2$ that depend on $M$ and satisfy and $c_1=c_2=0$ if and only if the Steenrod square $\mathop{\mathrm{Sq}}\nolimits^2$ acts trivially on $H^2(

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3: See Proposition \ref{['prop:kgrps']}
  • Corollary 1.4: See Proposition \ref{['prop:MS3']}
  • Lemma 2.1
  • proof
  • Lemma 2.2: cf. BH91
  • Lemma 2.3
  • proof
  • Lemma 2.4: cf. lipc2-an2
  • ...and 42 more