Manifold degree is functorial and detected in top cohomology

By [F2], $({\mathrm{id}}_M{)}_{\ast }[M]=[M]$, so uniqueness in [F1] gives degree one. For the composite, linearity and functoriality give $(g\circ f{)}_{\ast }[M]=g_{\ast }(\mathrm{deg}(f)[N])=\mathrm{deg}(f)\mathrm{deg}(g)[P]\mathrm{.}$ The coefficient is unique by [F1], proving the product formula in $\mathbb{Z}$. If either degree is zero, this same equality gives zero for the composite.

Negating the source fundamental class negates its image by [F2], hence its coordinate by [F1]. If the target class is replaced by $-[N]$, the same image $d[N]$ has coordinate $-d$ relative to the new class. Applying both changes gives $d$ again. These statements include $d=0$, where negation has no effect.

Now assume [F7]. A nonempty connected manifold is path-connected: the points reachable by a path from a fixed point form an open set, because each point has a path-connected coordinate-ball neighborhood and paths can be extended inside it. Every other reachability class is open for the same reason, so the first class has open complement. Connectedness forces it to be the whole manifold. Thus [F5] identifies $H_0(M;\mathbb{Z})$ with $\mathbb{Z}$ by total-coefficient augmentation. For an $n$-cocycle $\alpha$ and fundamental cycle $z={\sum }_i{m}_i{\sigma }_i$, [F6] gives $ϵ(\alpha \cap z)={\sum }_i{m}_i\alpha ({\sigma }_i)=\alpha (z)\mathrm{.}$ Hence [F4] followed by this augmentation is precisely the evaluation map $E_M:H^n(M;\mathbb{Z})\to \mathbb{Z}$, $a\mapsto ⟨a,[M]⟩$. It is an isomorphism, so $u_M=E_M^{-1}(1)$ exists uniquely and generates the entire group, not just a quotient modulo torsion. The same argument applies to $N$.

Let $\beta$ be a cocycle representing $u_N$ and $z$ a cycle representing $[M]$. By [F3], evaluation of $f^{\ast }{u}_N$ on $z$ is $(\beta f_{\mathrm{\#}})(z)=\beta (f_{\mathrm{\#}}z)$. The chain $f_{\mathrm{\#}}z$ represents $f_{\ast }[M]=\mathrm{deg}(f)[N]$ by [F1] and [F2]. Cocycle evaluation ignores a boundary since $\beta \mathrm{\partial }=0$, so this value is $\mathrm{deg}(f)⟨u_N,[N]⟩=\mathrm{deg}(f)$. Thus $E_M(f^{\ast }{u}_N)=\mathrm{deg}(f)=E_M(\mathrm{deg}(f)u_M)$. Injectivity of $E_M$ in step 1.3 proves the asserted equality. In particular degree zero gives zero pullback on the generator, and if the pullback is zero its evaluation makes the degree zero.

The degree identities are steps 1.1 and 1.2, and cohomology detection is step 2.1. For $n=0$, the manifolds are points, and [F1] gives degree ${ϵ}_M{ϵ}_N$; the normalized zero-cochain has value ${ϵ}_M$ on the source point, so pullback value ${ϵ}_N$ equals ${ϵ}_M{ϵ}_N{ϵ}_M$, verifying the formula explicitly. Empty manifolds are excluded by [F1]'s scalar-degree hypothesis. Coefficients are $\mathbb{Z}$, so zero-ring ambiguity does not arise. Degenerate simplices obey the evaluation computation in step 1.3. There is no AC in the identity, composition or orientation calculations; [F7] is used only through [F4] for the top-cohomology isomorphism.