First nonzero homotopy group of a sphere

If $n\ge 2$, apply [F4] with a singleton indexing set and its sphere based at $b$. This is the given sphere, not a wedge with an extra summand. It is a CW complex, is path connected, and has zero positive homotopy groups below $n$. Thus it satisfies exactly the $(n-1)$-connectivity hypothesis of [F1]. Assuming [A1], [F1] and [F2] give ${\pi }_n(S^n,b)\stackrel{h}{\to }{H}_n(S^n;\mathbb{Z})=\mathbb{Z}[S^n]$ as an isomorphism. This proves the asserted first nonzero group by the Hurewicz route.

For any based self-map $u$ of the oriented sphere, degree means the integer $d$ for which $u_{\ast }[S^n]=d[S^n]$, as used in [F3]. Consequently [F5] gives $h([u])=d[S^n]$. In particular $h([\mathrm{i}\mathrm{d}])=[S^n]$, and $h$ followed by the coefficient map $d[S^n]\mapsto d$ is exactly degree. This calculation also holds when $n=1$ because the defining formula in [F5] includes degree one.

For $n=1$, [F3] directly gives ${\pi }_1(S^1,b)\cong \mathbb{Z}$. There is no positive integer $i<1$, so the lower-vanishing assertion has no instance. Step 2.1 identifies the Hurewicz image of each degree class and in particular of the identity. For every $n\ge 2$, [F3] also supplies the degree isomorphism independently of [F1], while [F4] already gave lower vanishing choice-free. Thus the AC hypothesis belongs only to the indicated general-Hurewicz derivation, not to the independent sphere calculation.

The group in degree $n$ is nonzero because its identity-map class has coefficient $1$; the identity element of that group is instead the constant-map class with coefficient $0$. Negative coefficients are inverse classes by step 2.1. Empty spheres and $n=0$ are outside the hypothesis. Every specified basepoint is permitted by [F3] and the based singleton construction in [F4]; no family of basepoints or orientations is selected. This establishes the calculation and its orientation normalization in all stated degrees.