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Homotopy classification of based maps between $\mathbf{A}_n^2$-complexes

Pengcheng Li

Abstract

Let $X,Y$ be $(n-1)$-connected finite pointed CW-complexes of dimension at most $n+2$, $n\geq 3$. In this paper we give elementary proofs of the abelian group structure of $[X,Y]$ of homotopy classes of based maps from $X$ to $Y$, which was due to Baues and Schmidt. Furthermore, we determine the explicit generators associated to $[X,Y]$. As an application, we compute certain (sub)groups of self-homotopy equivalences of certain Chang complexes.

Homotopy classification of based maps between $\mathbf{A}_n^2$-complexes

Abstract

Let be -connected finite pointed CW-complexes of dimension at most , . In this paper we give elementary proofs of the abelian group structure of of homotopy classes of based maps from to , which was due to Baues and Schmidt. Furthermore, we determine the explicit generators associated to . As an application, we compute certain (sub)groups of self-homotopy equivalences of certain Chang complexes.

Paper Structure

This paper contains 9 sections, 19 theorems, 113 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and notation
  4. Homotopy cofibre sequences for indecomposable $\mathsf{A}_n^2$-complexes
  5. Groups $[X,Y]$ and their explicit generators
  6. Proofs of groups and generators in Tables \ref{['table1']},\ref{['table2']}
  7. Self-homotopy equivalences of Chang complexes
  8. Proof of Theorem \ref{['thm:ECC']}
  9. Proofs of Theorems \ref{['thm:LDU']}, \ref{['thm:Ecc-H']} and Corollary \ref{['cor:Ecc:n+2']}

Key Result

Theorem 1.1

Let $n\geq 3$ and let $r,t\geq 1$ be integers. There is a split short exact sequence \begin{tikzcd}[column sep=small] \zz{\min(r,t)+1}\ar[r,tail]&\E(\CC)\ar[r,two heads,"\phi"]&\Aut(\zz{t+1})\oplus\Aut(\zz{r+1}), \end{tikzcd}where $\min(r,t)$ denotes the minimum of $r$ and $t$, $\phi(f)=(\pi_{ where $p\colon \mathbb{Z}/2^{t+1}\to\mathbb{Z}/2^{\min(r,t)+1}$ and $q\colon \mathbb{Z}/2^{r+1}\to

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 25 more