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The unstable homotopy groups of 2-cell complexes

Zhongjian Zhu

Abstract

In this paper, we develop the new method, initiated by B. Gray (1972), to compute the unstable homotopy groups of the mapping cone, especially for $2$-cell complex $X=S^m\cup_α e^{n}$. By Gray's work mentioned above or the traditional method given by I.M.James (1957) which were widely used in previous related work to compute $π_{i}(X)$, the dimension $i\leq 2n+m-4$. By our method, we can compute $π_{i}(X)$ for $i>2n+m-4$. We use this different technique to generalize J.Wu's work, at Mem. of AMS, on homotopy groups of mod $2$ Moore spaces to higher dimensional homotopy groups of mod $2^r$ Moore spaces $P^{n}(2^r)$ for all $r\geq 1$. This practice shows that the technique given here is a new general method to compute the unstable homotopy groups of CW complexes with higher dimension.

The unstable homotopy groups of 2-cell complexes

Abstract

In this paper, we develop the new method, initiated by B. Gray (1972), to compute the unstable homotopy groups of the mapping cone, especially for -cell complex . By Gray's work mentioned above or the traditional method given by I.M.James (1957) which were widely used in previous related work to compute , the dimension . By our method, we can compute for . We use this different technique to generalize J.Wu's work, at Mem. of AMS, on homotopy groups of mod Moore spaces to higher dimensional homotopy groups of mod Moore spaces for all . This practice shows that the technique given here is a new general method to compute the unstable homotopy groups of CW complexes with higher dimension.

Paper Structure

This paper contains 16 sections, 46 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. Higher order Whitehead products and relative James construction
  3. Higher order Whitehead products
  4. Relative James construction
  5. homotopy of $\gamma_n$ and Hopf invariant
  6. homotopy of $\Omega\Sigma X\xrightarrow{\partial} F_p$ and extension problem
  7. Skeletons of the homotopy fiber
  8. Homotopy groups of $4,5$ and $6$ dimensional mod $2^r$ Moore spaces
  9. Homotopy groups of $P^{4}(2^r)$
  10. Homotopy groups of $P^{5}(2^r)$
  11. Homotopy groups of $P^{6}(2^r)$
  12. Homotopy groups of $7$ and $8$ dimensional mod $2^r$ Moore spaces
  13. Homotopy groups of $P^{7}(2^r)$
  14. Homotopy groups of $P^{8}(2^r)$
  15. Some remarks
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X\xrightarrow{f}Y$ be a map of simply connected CW complexes, $X=\Sigma X'$, $Y=\Sigma Y'$. Then $J_n(M_f,X)$ has the homotopy type $Y\cup_{\gamma_2}C(\Sigma Y'\wedge X')\cup_{\gamma_3}\cdots \cup_{\gamma_n}C(\Sigma^{n-1} Y'\wedge X'^{\wedge{n-1}})$, $\gamma_r$ is an element of $r$-th order Whi

Theorems & Definitions (87)

  • Theorem 1.1: Theorem 3.4 of ZJ
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5: Lemma 2.2 of ZP23
  • Lemma 2.6
  • ...and 77 more