Regular-value formula for degree

The first formula and finiteness of $S$ are [F1]. By [F2], the supplied smooth orientations give compatible integral local generators ${\mu }_p$ on $M$ and ${\nu }_q$ on $N$. Thus [F3] supplies fundamental classes and [F4] supplies a unique integer $d$ with $F_{\ast }[M]=d[N]$. We will compute $d$ by restriction to the stalk at the specified value $y$, without any comparison of de Rham representatives with unspecified Kronecker pairings.

First let $S=\{p_1,\dots ,p_m\}$ be nonempty. Choose pairwise disjoint open neighbourhoods $U_i$ of these finitely many points. Each can be small enough for the germ calculation in [F2] and contains no other point of $S$. Finite Hausdorff separations provide disjointness: for each distinct pair choose disjoint neighbourhoods and intersect the finitely many associated ones at each point. Let $U={\bigcup }_i{U}_i$. The closed set $Z=M\setminus U$ lies in $M\setminus S$, which is open since a finite set is closed in a Hausdorff manifold. Therefore [F5] gives $H_n(M,M\setminus S;\mathbb{Z})\cong H_n(U,U\setminus S;\mathbb{Z})\cong {⨁}_{i=1}^m{H}_n(U_i,U_i\setminus \{p_i\};\mathbb{Z}),$ the second isomorphism being [F6]. The coordinate projections of this identification agree, after local excision, with restriction to $H_n(M,M\setminus \{p_i\})$: on the $i$th summand this is inclusion, and every other summand lies wholly in the subspace $M\setminus \{p_i\}$ and therefore is zero in that quotient.

If $S=\mathrm{\varnothing }$, the map lands in $N\setminus \{y\}$. Its induced chain map becomes zero after quotient by that subspace. Consequently the restriction of $F_{\ast }[M]=d[N]$ at $y$ is zero, hence $d{\nu }_y=0$ and $d=0$. This equals the compact-support degree and empty signed sum in [F1]. This argument does not require the punctured target to be contractible.

Restrict $[M]$ to the group in step 2.1. By the defining local restrictions [F3] and the coordinate identification just proved, its components are exactly $({\mu }_{p_1},\dots ,{\mu }_{p_m})$. Because $F(M\setminus S)\subset N\setminus \{y\}$, it gives a map of pairs $(M,M\setminus S)⟶(N,N\setminus \{y\})\mathrm{.}$ On the $i$th summand its action is the local germ action from [F2], sending ${\mu }_{p_i}$ to ${\epsilon }_i{\nu }_y$, where ${\epsilon }_i=\mathrm{sgn}(d{F}_{p_i})$. Additivity and [F7] therefore send the restricted fundamental class to $({\sum }_i{\epsilon }_i){\nu }_y$.

The alternative route is to first apply $F_{\ast }$ to $[M]$ and then restrict at $y$. By [F7] these routes agree, since both are induced by $F_{\mathrm{\#}}$ followed by the quotient by chains in $N\setminus \{y\}$. The first route gives $d{\nu }_y$ by [F3], [F4] and step 1.1; step 3.1 gives the other. The element ${\nu }_y$ is a generator of an infinite cyclic group by [F2], so $d{\nu }_y=\left({\sum }_i{\epsilon }_i\right){\nu }_y⟹d={\sum }_i{\epsilon }_i\mathrm{.}$ Together with [F1] this proves equality of homological and compact-support degrees.

At $n=0$, each nonempty connected manifold is a point. By [F2] and [F4], $F_{\ast }({\epsilon }_M[p])={\epsilon }_M[q]=({\epsilon }_M{\epsilon }_N){\epsilon }_N[q]$, so the homological coefficient equals the local ray sign and the compact-support degree in [F1]. At $n=1$ step 2.1 is an ordinary degree-one relative group calculation and the local sign computation in [F2] already uses the reduced $H_0$ difference of the two sides of a point. A singleton fibre and cancellation to degree zero are included in step 4.1. All maps are on unnormalized quotient chains, so no degenerate-simplex exception occurs. Only finitely many disjoint neighbourhoods for the given finite fibre are selected; [F1]–[F7] use no AC in these clauses. The common Euclidean orientation convention matters: negating it reverses both fundamental classes and leaves the homological coefficient unchanged.