Degree-d sphere maps act by multiplication by d in any generalized theory

All maps used are in the functors' domains. For $p\ge 1$ the skeleton of $S^p$ in each dimension $0\le k<p$ is its basepoint and in dimensions $k\ge p$ is the entire sphere. The same description holds for a finite wedge with its common vertex. Every based continuous map among these spaces therefore preserves every skeleton and is cellular. For $p=0$ every map of the discrete vertex sets is cellular. In particular this applies to $f$, pinch, fold, wedge inclusions, collapses and wedge maps; the homotopy axiom applies to based homotopies between these cellular endpoints. Let $r$ be the geometric degree in [F1]. Then $[f]=r[\mathrm{i}\mathrm{d}]$ under pinch addition. We compare $r$ with the homological degree $d$ below.

Write $P:S^p\to S^p\vee S^p$ for the pinch and $F:S^p\vee S^p\to S^p$ for the fold, so that for based self-maps $\alpha ,\beta$ the pinch sum is represented by $F\circ (\alpha \vee \beta )\circ P$; the two components ${\pi }_{1}P,{\pi }_{2}P$ are based degree-one maps, hence are based homotopic to the identity.

For $p=0$ the only based self-maps are identity and collapse. Identity induces identity by functoriality; collapse factors through a point, whose reduced groups are zero by the empty-wedge clauses of [F3] and [F4], so it induces zero. These are also the integers acting on reduced $H_0(S^0;\mathbb{Z})$ by [F6]. No degree is defined here using unreduced $H_0(S^0;\mathbb{Z})\cong {\mathbb{Z}}^2$.

For homology, $P_{\ast }:{\tilde{h}}_n(S^p)\to {\tilde{h}}_n(S^p\vee S^p)$ has components $({\pi }_{1}P{)}_{\ast }=({\pi }_{2}P{)}_{\ast }=\mathrm{i}\mathrm{d}$ under the wedge isomorphism of [F3], hence is the diagonal; and $F_{\ast }$ is the sum map because $F$ restricts to the identity on each summand. Therefore $(F\circ (\alpha \vee \beta )\circ P{)}_{\ast }={\alpha }_{\ast }+{\beta }_{\ast }$ for all based $\alpha ,\beta$.

For cohomology, $F^{\ast }$ is the diagonal and $P^{\ast }$ is the sum map under the wedge isomorphism of [F4], because $F$ restricts to the identity on each summand and the components of $P$ are degree one; therefore $(F\circ (\alpha \vee \beta )\circ P{)}^{\ast }={\alpha }^{\ast }+{\beta }^{\ast }$ for all based $\alpha ,\beta$.

By step 1.1 and functoriality of $\tilde{h}$, $f_{\ast }$ equals the $r$-fold sum of ${\mathrm{i}\mathrm{d}}_{\ast }$ in the group $\mathrm{End}({\tilde{h}}_n(S^p))$, which is multiplication by $r$ by step 2.1. For negative $r$, an inverse class has the negative induced map, since adding it to the original class gives a nullhomotopic map, which factors through the zero reduced group of a point. The same observation includes $r=0$.

By step 1.1 and functoriality of $\tilde{g}$, $f^{\ast }$ equals the $r$-fold sum of ${\mathrm{i}\mathrm{d}}^{\ast }$ in the group $\mathrm{End}({\tilde{g}}^n(S^p))$, which is multiplication by $r$ by step 2.2, including $r<0$: the additive inverse class induces the negative endomorphism since its pinch sum with the original is null and hence induces zero through a point.

The same finite pinch calculation in ordinary integral homology identifies the two degree conventions. By [F6], the top homology of a finite wedge of $p$-spheres is free on the sphere inclusions: its cellular complex has a common vertex, one generator per $p$-cell, and zero boundary (also for $p=1$, since both endpoints attach to that vertex). The collapses give the corresponding coordinate projections. Thus pinch is diagonal and fold is addition in top homology, exactly as in step 2.1. Homotopy invariance [F6] and $[f]=r[\mathrm{i}\mathrm{d}]$ give $f_{\ast }[S^p]=r[S^p]$. By the definition [F2], $d=r$.

Combining steps 1.3, 3.1 and 3.2 gives the asserted multiplication by the degree on every reduced homology and cohomology group of $S^p$.