Cap duality on a Euclidean coordinate ball

Let $K_j=\{x:\parallel x\parallel \le j\}$ for integers $j\ge 1$. Every compact subset lies in some $K_j$ by [F2], so these supports are cofinal. The complement of $K_j$ retracts to the sphere of radius $j+1$ by the radial homotopy with norm $(1-t)\parallel x\parallel +t(j+1)>j$. The ambient space is contractible. The pair sequence [F4] thus gives $H_i(U,U\setminus K_j;\mathbb{Z})\cong \{\begin{array}{ll}\mathbb{Z},& i=n,\\ 0,& i\mathrm{\ne }n\mathrm{.}\end{array}$ At $i=1$ the pair sequence uses the kernel of the augmentation on the complement, and at $i=0$ it gives zero because the complement is nonempty. Thus $n=1$, with two complement components, is included. Negative groups are zero.

Apply relative UCT [F5] with coefficient group the additive group of $R$. Every integral relative homology group in step 1.1 is either $0$ or $\mathbb{Z}$. Their ${\mathrm{Ext}}^1$ terms vanish: use the zero resolution for $0$ and the length-zero free resolution $\mathbb{Z}\stackrel{1}{\to }\mathbb{Z}$ for $\mathbb{Z}$ (there is no positive resolution term, so the degree-one Hom cohomology is zero). Consequently evaluation gives $H^p(U,U\setminus K_j;R)\cong {\mathrm{Hom}}_{\mathbb{Z}}(H_p(U,U\setminus K_j;\mathbb{Z}),R)\mathrm{.}$ This is $R$ for $p=n$ and zero otherwise. The isomorphism is evaluation on integral relative cycles, not an unspecified additive isomorphism. The invocation of UCT uses AC for arbitrary-rank cycle/boundary freeness, projections and free comparison lifts in its proof; no additional choice enters this calculation.

Choose one integral generator $e_0$ of the local stalk at the center. By [F3] there is a unique integral class $e_j$ supported on $K_j$ restricting to $e_0$. Its coefficient extension ${\bar{e}}_j$ is an $R$-module generator. Indeed, for $n>1$ the radial pair calculation in [F4] sends it to the corresponding reduced sphere generator; for $n=1$ it sends it to the difference of the two point generators in the augmentation kernel of $H_0(S^0)$, not to either point generator separately. Replacing integral coefficients by their images in $R$ gives the respective generator over $R$ in both cases. At the center the given $R$-orientation is $u{\bar{e}}_0$ for a unit $u\in R$: writing ${\bar{e}}_0=v(u{\bar{e}}_0)$ because the orientation generates gives $vu=1$. Point restriction is injective, so $[U{]}_{K_j}=u{\bar{e}}_j$ for every $j$. Restriction sends $e_{j+1}$ to $e_j$ since both have center value $e_0$.

Naturality of evaluation in [F5] shows that the support transition in degree $n$ has the same coordinate in $R$: evaluating the transitioned class on $e_{j+1}$ equals evaluating the original class on its restriction $e_j$. Thus under step 2.1 every transition is the identity of $R$ in degree $n$. In other degrees all groups are zero. The colimit [F2] is therefore $R$ in degree $n$ and zero in all other degrees.

If $a\in H^n(U,U\setminus K_j;R)$ evaluates to $r\in R$ on $e_j$, choose an integral relative cycle $z_j$ representing $e_j$ and a relative cocycle $\phi$ representing $a$. Then [F1] represents its cap image by $\phi \cap (u{\bar{z}}_j)$. In degree $n$ the cap formula retains the last vertex of each $n$-simplex. Applying zero-chain augmentation therefore gives $ϵ(\phi \cap (u{\bar{z}}_j))=u\phi (z_j)=ur\mathrm{.}$ This chain is an absolute cycle and its class is independent of representatives by [F1]. Since $U$ is contractible, [F4] identifies $H_0(U;R)$ with $R$ via augmentation: the map to a point is a homotopy equivalence, and the point complex has $H_0=R$. Thus $D_U$ is multiplication by the unit $u$ in these coordinates. In particular the class with $r=u^{-1}$ evaluates to $1$ on the oriented class and maps to the positive point generator.

Steps 3.1 and 3.2 prove the isomorphism in degree $n$. If $p\mathrm{\ne }n$, the source is zero by step 3.1; the target is zero because a contractible space has zero homology in positive degrees, and negative chain degrees are zero. When $n=0$, $U$ is a point and is itself terminal compact support. Its integral point complex and its cochain complex give $H^0(U;R)=H_0(U;R)=R$ and zero in other degrees: their positive differentials alternate between identity and zero. The orientation is a unit $u$ times the point class, and degree-zero cap again multiplies by $u$. Thus the same conclusion holds.

Over the zero ring all displayed modules and maps are zero, with the unique unit satisfying $1=0$, and the isomorphism statement still holds. An open ball is nonempty by hypothesis; empty supports in its colimit contribute only zero. The radial retractions in step 1.1 preserve the strict complement even at $t=0,1$. The cap evaluation includes all unnormalized simplex generators. Only one base generator and finitely many representatives for an individual calculation were used beyond the stated AC in step 2.1.