Coordinate-ball classes identify local homology stalks

The ball $K$ is compact by [F6]; compactness of its chart preimage follows by pulling any open cover back to the Euclidean ball and taking a finite subcover there. It is closed in Hausdorff $M$. Thus $Z=M\setminus B$ is closed and contained in the open set $M\setminus K$. Excision gives $H_n(M,M\setminus K;R)\cong H_n(B,B\setminus K;R)$. It gives the analogous isomorphism for $\{x\}$, and these identifications commute with the restriction maps since all are induced by inclusions.

Put $\rho =(r+s)\mathrm{/}2$. The annulus $B\setminus K$ retracts onto $S_{\rho }=\{u:\parallel u\parallel =\rho \}$ by $u\mapsto (1-t+t\rho \mathrm{/}\parallel u\parallel )u$: throughout the homotopy its norm is strictly between $r$ and $s$. The inclusion $S_{\rho }\to {\mathbb{R}}^n\setminus \{x\}$ is a homotopy equivalence on homology. Indeed translate the target by $-x$; the resulting map $u\mapsto u-x$ is homotopic to $u\mapsto u$ by $u\mapsto u-tx$, which never vanishes because $\parallel tx\parallel <r<\rho$. The centered sphere is a radial deformation retract of ${\mathbb{R}}^n\setminus \{0\}$. Hence inclusion $B\setminus K\to {\mathbb{R}}^n\setminus \{x\}$ induces isomorphisms on reduced homology in all degrees, including the augmentation kernel in degree zero.

Choose $0<ϵ<s-\parallel x\parallel$. Both $B\setminus \{x\}$ and ${\mathbb{R}}^n\setminus \{x\}$ radially retract about $x$ to the sphere of radius $ϵ$: $h_x(y,t)=x+(1-t+\frac{tϵ}{\parallel y-x\parallel })(y-x)\mathrm{.}$ For the first space the homotopy lies in $B$, since it is the segment between $y$ and a point of that small sphere, both in the convex ball $B$. It avoids $x$ by positivity of the radial coefficient. These retractions commute with inclusion, so that inclusion induces isomorphisms. Factoring the isomorphism of step 1.2 through $B\setminus \{x\}$ proves that $B\setminus K\to B\setminus \{x\}$ induces reduced homology isomorphisms.

The ball $B$ is contractible. As calculated in [F1], the exact sequence [F3] identifies $H_n(B,A;R)$ with ${\tilde{H}}_{n-1}(A;R)$ for either nonempty subspace $A=B\setminus K$ or $B\setminus \{x\}$. At $n=1$ this is the kernel of $H_0(A;R)\to H_0(B;R)$; no unreduced degree-zero replacement is made. Naturality of the connecting map and step 2.1 therefore prove the desired restriction isomorphism, after step 1.1. Its target is $R$ by [F1].

If $J\subset \mathrm{int}K$, choose any $y\in \mathrm{int}J$. Restriction from $K$ to $y$ factors as restriction from $K$ to $J$ followed by restriction from $J$ to $y$, by [F5]. Both the composite and the latter map are isomorphisms by step 3.1 applied to the respective charts; hence the first map is an isomorphism. The same factorization holds for every $y$ in the interior of $J$, proving compatibility without any common-coordinate assumption on the two balls. For $n=0$ all these balls are singletons; excision reduces their groups and maps to the point group $R$ and its identity, as in [F1].

The statement requires an actual chart ball, so supplies no ball in an empty manifold. Zero coefficients give isomorphisms of zero modules. The radii have strict inequalities; neither radius zero nor a point on $\mathrm{\partial }K$ is claimed. All homotopies include $t=0,1$ and operate on unnormalized singular chains through [F4]. Only finitely many individual radii, points and charts are used; all comparison maps are canonical inclusions, and no AC is invoked.