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The $n+3$, $n+4$ dimensional homotopy groups of $\mathbf{A}_n^2$-complexes

Tian Jin, Zhongjian Zhu

Abstract

In this paper, we calculate the $n+3$, $n+4$ dimensional homotopy groups of indecomposable $\mathbf{A}_n^2$-complexes after localization at 2. This job is seen as a sequel to P.J. Hilton's work on the $n+1,n+2$ dimensional homotopy groups of $\mathbf{A}_n^2$-complexes (1950-1951). The main technique used is analysing the homotopy property of $J(X,A)$, defined by B. Gray for a CW-pair $(X,A)$, which is homotopy equivalent to the homotopy fibre of the pinch map $X\cup CA\rightarrow ΣA$. By the way, the results of these homotopy groups have been used to make progress on recent popular topic about the homotopy decomposition of the (multiple) suspension of oriented closed manifolds.

The $n+3$, $n+4$ dimensional homotopy groups of $\mathbf{A}_n^2$-complexes

Abstract

In this paper, we calculate the , dimensional homotopy groups of indecomposable -complexes after localization at 2. This job is seen as a sequel to P.J. Hilton's work on the dimensional homotopy groups of -complexes (1950-1951). The main technique used is analysing the homotopy property of , defined by B. Gray for a CW-pair , which is homotopy equivalent to the homotopy fibre of the pinch map . By the way, the results of these homotopy groups have been used to make progress on recent popular topic about the homotopy decomposition of the (multiple) suspension of oriented closed manifolds.
Paper Structure (6 sections, 15 theorems, 86 equations)

This paper contains 6 sections, 15 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Some notations and lemmas
  3. Elementary Moore spaces
  4. Elementary Chang-complexes
  5. The $n+3$- homotopy groups of elementary Chang-complexes
  6. The $n+4$-homotopy groups of elementary Chang-complexes

Key Result

Theorem 1.1

The $n+3,n+4$-homotopy groups of all $2$-local noncontractible indecomposable $\mathbf{A}_n^2$-complexes (except spheres) are listed as follows: In the above tables, an integer $n$ indicates a cyclic group $\mathbb{Z}_n:=\mathbb{Z}/n\mathbb{Z}$ of order $n$, the symbol $\infty$ the group $\mathbb{Z}_{(2)}$ (the $2$-local integers), the symbol $"+"$ the direct sum of the groups and $(2)^k$ indicat

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4: Theorem 1.16 of Cohen
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 17 more