A collar constructs the relative orientation class and its boundary class

Fix the collar from [F1]. For $0<d<1$ put $C_d=c(A\times [0,d))$ and $K_d=M\setminus C_d$. Then $K_d$ is a compact subset of $N$. The inclusion $A\to C_d$ is a deformation retract, by reducing collar height linearly to zero, so [F4] and the natural pair sequences [F3] give an isomorphism $H_i(M,A;R)\to H_i(M,C_d;R)$. Naturality here follows directly from the inclusions and quotient maps on the two pair short exact chain sequences. Excision of the closed set $A$, contained in the open $C_d$, then gives an isomorphism $H_i(N,N\setminus K_d;R)\stackrel{\sim }{\to }{H}_i(M,C_d;R)\mathrm{.}$ Hence there is a unique $z_d\in H_n(M,A;R)$ corresponding to the orientation class $[N{]}_{K_d}$ of [F2].

If $0<d^{\mathrm{\prime }}<d$, then $K_d\subset K_{d^{\mathrm{\prime }}}$. The support restriction sends $[N{]}_{K_{d^{\mathrm{\prime }}}}$ to $[N{]}_{K_d}$ by [F2], and all maps in step 1.1 commute with these inclusions since they come from chain quotient maps. Thus $z_{d^{\mathrm{\prime }}}=z_d$. Write $z$ for their common value. Every $x\in N$ lies in a $K_d$: outside the collar this is immediate, and at positive collar height choose $d$ less than that height. The restriction of $z$ at $x$ is consequently the prescribed generator, using step 1.1 for that $d$. Conversely, any class with these interior restrictions has image $[N{]}_{K_d}$ by [F2]'s pointwise injectivity for one fixed $K_d$, and the isomorphism in step 1.1 makes it equal to $z$. This proves both existence and uniqueness even if $M$ has closed components.

Fix $x\in A$, numbers $0<d<e<1$, and the compact vertical fiber $J=c(\{x\}\times [0,e])$. Put $L=M\setminus J$. The relative groups $H_i(M,L;R)$ vanish in every degree. To see this, restrict by [F3] to a boundary chart in the collar containing $J$, identified with ${\mathbb{R}}^{n-1}\times [0,\mathrm{\infty })$ by reparametrizing height in a slightly longer collar segment. The fiber becomes $\{0\}\times [0,E]$ for some $E>0$. The half-space is contractible. Its complement of this fiber is also contractible: first push every height upward by $E+1$, a homotopy avoiding the fiber because a point on the vertical axis starts above $E$ and a point off it stays off it; then linearly contract to $(0,E+1)$ in the region of heights greater than $E$. The pair exact sequence [F3] and [F4] give zero relative homology, including degree zero since both spaces are nonempty and connected. This also proves the assertion for $n=1$.

Write $C=C_{\ast }(M;R)$, $D=C_{\ast }(C_d;R)$ and $E=C_{\ast }(L;R)$. There is a short exact sequence $0\to D\mathrm{/}(D\cap E)⟶C\mathrm{/}E⟶C\mathrm{/}(D+E)\to 0.$ The first map sends a $D$ chain to its class modulo $E$, with kernel $D\cap E$; the last quotient is onto and its kernel is the image of $D$. The simplex basis identifies $D\cap E=C_{\ast }(C_d\cap L;R)$. Since $C_d,L$ are open, the explicit comparison in [F5] identifies $H_{\ast }(C\mathrm{/}(D+E))$ with $H_{\ast }(M,C_d\cup L;R)$. The latter is the group supported on the interior segment $J^{\mathrm{\prime }}=J\setminus C_d=c(\{x\}\times [d,e])\mathrm{.}$ By step 2.2 the middle complex has zero homology. The connecting map of [F5] is therefore an isomorphism $H_n(M,M\setminus J^{\mathrm{\prime }};R)\stackrel{\sim }{\to }{H}_{n-1}(C_d,C_d\cap L;R)\mathrm{.}$ The collar retraction is a homotopy equivalence of pairs from $(C_d,C_d\cap L)$ to $(A,A\setminus \{x\})$: at every retained point the base coordinate differs from $x$, and reducing height never changes it.

The group on the left of step 3.1 is free of rank one, and the restriction of $z$ to it is a generator. Indeed excision identifies it with the group supported on a compact straight segment in an interior coordinate ${\mathbb{R}}^n$. For a point $y$ of that segment, radially retract both its complement and the punctured space about $y$ to a sphere of radius larger than the segment. Outward radial motion from a point outside a convex segment cannot enter it: if a farther radial point belonged to the segment, convexity with $y$ would put the original point in it. Motion inward from outside the large sphere remains outside the segment. Thus inclusion of these complements is a homotopy equivalence, and their pair sequences identify restriction with the local group at $y$, isomorphic to $R$. By step 2.1 the restriction of $z$ there is the prescribed generator. This proves the assertion without assuming that an arbitrary compact support has cyclic top homology.

The image of $\mathrm{\partial }z$ in $H_{n-1}(A,A\setminus \{x\};R)$ is exactly the image of $z$ under the isomorphism in step 3.1 followed by collar retraction. This is naturality for explicit short exact complexes: map $C_{\ast }(A)$ to $D\mathrm{/}(D\cap E)$ by inclusion and quotient, map $C$ to $C\mathrm{/}E$, and map $C\mathrm{/}{C}_{\ast }(A)$ to $C\mathrm{/}(D+E)$. The maps are well-defined since $A\subset C_d$; both squares commute. A relative cycle representing $z$ lifts to the same chain in the two middle complexes, so taking its boundary gives exactly the two stated connectors. The quotient comparison in [F5] is induced by the quotient map and hence preserves this interpretation. Therefore step 4.1 and the isomorphisms of step 3.1 show that the restriction of $\mathrm{\partial }z$ at every $x$ is a generator.

These local values form a continuous orientation section on $A$. For any sufficiently small closed coordinate ball in $A$, first restrict the single global class $\mathrm{\partial }z$ to its support-relative group; its point restrictions are the basic local-system section of [F2] in dimension $n-1$. Hence the pointwise assignment is continuous locally and thus globally. The boundary is compact by [F1], so [F7] identifies $\mathrm{\partial }z$ with the fundamental class of this orientation. No family of local generators has been selected: the one class $\mathrm{\partial }z$ determines all of them.

To compute the sign at $x$, choose a small oriented affine $(n-1)$-simplex $T$ in a boundary chart, with $x$ in its interior, and a collar interval $[0,e^{\mathrm{\prime }}]$ with $e<e^{\mathrm{\prime }}<1$ lying in that chart. Let $t$ be its positively directed singular one-simplex. The shuffle chain $w=T\times t$ is a relative cycle modulo $C_d\cup L$: its side faces miss the vertical fiber, its top has height above $e$, and its bottom lies in $C_d$. It is a generator of the left group of step 3.1. To verify this, choose a height in $(d,e)$ avoiding the finitely many internal faces of the shuffle triangulation along the vertical line through the interior point $x$. Indeed the staircase shuffle faces have height equal to one of the finitely many partial sums of the barycentric coordinates of $x$, multiplied by $e^{\mathrm{\prime }}$; all these coordinates are positive since $x$ is interior. Thus no internal face contains a vertical interval, and vertical side faces miss the line. Near the resulting point the shuffle chain is one affine top simplex with coefficient $+1$ or $-1$, and excision and the oriented-simplex calculation in [F7] make its local class a generator. Step 4.1's restriction isomorphism then makes $[w]$ a generator. This is the product orientation corresponding to tangent coordinates followed by the inward coordinate. Write the given orientation class there as $u[w]$, where $u$ is a unit of $R$. The boundary formula [F6] gives $\mathrm{\partial }w=(\mathrm{\partial }T)\times t+(-1{)}^{n-1}T\times ([e^{\mathrm{\prime }}]-[0])\mathrm{.}$ In the connecting target modulo $L$, only the bottom remains, with coefficient $(-1{)}^n$. Thus the induced boundary generator is $(-1{)}^{n}u[T]$. Moving an inward last coordinate past $n-1$ tangent coordinates contributes $(-1{)}^{n-1}$ in the signed shuffle convention, and replacing inward by outward contributes $-1$; their product is $(-1{)}^n$. This is exactly outward-normal-first.

For $n=1$, $T$ is a point, the side term is zero, and $\mathrm{\partial }[0,e^{\mathrm{\prime }}]=[e^{\mathrm{\prime }}]-[0]$ gives the same negative bottom sign. Empty $M$ or empty boundary was handled in the Given paragraph, and the zero ring gives the unique generator of each zero module and the zero class throughout. Closed components are contained in every $K_d$ and were included in the uniqueness argument. Degenerate singular simplices are retained in all quotient complexes; the particular prism used for the sign is an explicit affine triangulation. The endpoints $0<d<e<e^{\mathrm{\prime }}<1$ ensure the support segment is interior and its upper prism face is outside the fiber. Every construction uses a supplied collar, compact-support uniqueness, a single chosen point or finite simplex data. No AC or simultaneous local-generator selection occurs.