The unstable homotopy groups of 2-cell complexes

TL;DR

This paper introduces a new method to compute the unstable homotopy groups of 2-cell complexes, extending previous techniques to higher dimensions and generalizing results for mod 2 Moore spaces.

Contribution

The paper develops a novel technique for calculating unstable homotopy groups of CW complexes, surpassing traditional methods in higher dimensions and applying to mod 2^r Moore spaces.

Findings

Extended computation of homotopy groups beyond previous dimension limits

Generalized Wu's results to higher-dimensional mod 2^r Moore spaces

Demonstrated the method's effectiveness for complex CW structures

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 $X=S^m\cup_{\alpha} e^{n}$. 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 $\pi_{i}(X)$, the dimension $i\leq 2n+m-4$. By our method, we can compute $\pi_{i}(X)$ 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.