Canonical twisted fundamental classes over compact subsets

The local element $u_x$ is canonically typed and varies as a section. [F1] On a coordinate ball $B$ containing $x$, choose a local integral orientation generator $o$. Trivialize ${\mathcal{O}}_M^R{\mathrm{\mid }}_B$ by $o\otimes 1_R$. A relative cycle for $o_x$ carrying coefficient $o_x\otimes 1_R$ then represents the element written $o_x\otimes (o_x\otimes 1_R)$ and corresponds to $1_R$. Replacing $o$ by $-o$ reverses the relative cycle and negates its coefficient, so the class is unchanged. Transport changes both factors by the same sign. Thus the classes $u_x$ are independent of the trivialization and form the constant coefficient-one section in every orientation chart.

Local chains give the closed-support Mayer--Vietoris sequence needed below. [F2, F4, given] For compact $C,D\subseteq M$, put $U=M\setminus C$, $V=M\setminus D$, and let $Q$ be the intrinsic local chain complex of $M$. With $Q_U=C_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(U;{\mathcal{O}}_M^R{\mathrm{\mid }}_U)$ and similarly for $V$, the same quotient calculation as the exact pattern in [F2] gives $0⟶Q\mathrm{/}(Q_U\cap Q_V)⟶Q\mathrm{/}{Q}_U\oplus Q\mathrm{/}{Q}_V⟶Q\mathrm{/}(Q_U+Q_V)⟶0,$ where the first map is diagonal and the second is difference. The common simplex generators give $Q_U\cap Q_V=C_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(U\cap V)$. The local subdivision and prism comparison in [F2] identifies the last quotient with the relative complex for $U\cup V$; coefficient transports along affine subpaths satisfy the same face cancellations. Since compact subsets of the Hausdorff manifold are closed by [F4], $U,V$ are open. The resulting long exact sequence is $\cdots \to H_{q+1}(M\mathrm{\parallel }C\cap D)\to H_q(M\mathrm{\parallel }C\cup D)\to H_q(M\mathrm{\parallel }C)\oplus H_q(M\mathrm{\parallel }D)\to H_q(M\mathrm{\parallel }C\cap D)\to \cdots ,$ where $H_q(M\mathrm{\parallel }E)=H_q(M,M\setminus E;{\mathcal{O}}_M^R)$. All maps are the support restrictions just displayed; no splitting is chosen.

Every compact convex coordinate support has vanishing, pointwise injectivity, and its canonical class. [F1, F2, F3, F4, step 1.1] Let $C$ be nonempty and compact convex in a coordinate chart identified with ${\mathbb{R}}^n$, and fix $x\in C$. Choose $L>{\sup }_{a\in C}\parallel a-x\parallel$. Radially move each point of ${\mathbb{R}}^n\setminus C$ to the sphere of radius $L$ about $x$. If its initial radius is at most $L$, the ray cannot meet $C$ farther out, since convexity with $x\in C$ would put the initial point in $C$; if the radius is larger, the whole motion stays outside the radius-$L$ ball. The same formula retracts ${\mathbb{R}}^n\setminus \{x\}$ to that sphere. Excision [F2], the pair sequences and homotopy invariance in [F3], and the point-local computation [F4] therefore identify $H_i(M\mathrm{\parallel }C)$ with the point-local group at $x$. It vanishes above $n$, and restriction in degree $n$ is injective. The inverse image of $u_x$ restricts to every $u_y$ by the constant section of step 1.1, so it is the unique canonical class. For $n=0$, a nonempty convex coordinate support is a point and the comparison is literal; the empty support has its unique zero class.

The three conclusions extend to finite unions of convex compact sets in one chart. [step 1.2, step 2.1] Induct on the number of sets. On adjoining the last convex set, its intersection with the preceding union is a union of fewer compact convex sets, since pairwise intersections remain convex or empty. Step 1.2 and vanishing above $n$ make restriction from the union injective. The two canonical classes agree on the intersection by pointwise injectivity there, so exactness glues them. Pointwise injectivity on the two pieces proves uniqueness, and the same exact window proves vanishing above $n$.

The finite-chain enlargement proves the three conclusions for every compact support lying in one chart. [F2, F4, step 2.1, step 3.1] Let $K$ be such a compact support and let $\alpha \in H_i(M\mathrm{\parallel }K)$ for $i\ge n$. By excision and the finite-chain definition in [F2], represent it in the coordinate space by a finite local chain $z$ whose boundary is supported outside $K$. The union $C$ of the images of the finitely many simplices occurring in $\mathrm{\partial }z$ is compact: each standard simplex is closed and bounded by [F4], its continuous image is compact, and a finite union of compact sets is compact. It is closed in the Hausdorff manifold and disjoint from $K$. Closed coordinate balls centered at points of $K$ and small enough to miss $C$ have interiors covering $K$; take a finite subcover and call its union $D$. Then $D$ is a finite union of convex compact sets, and $z$ represents a class ${\alpha }_D\in H_i(M\mathrm{\parallel }D)$ restricting to $\alpha$. If $i>n$, step 3.1 makes ${\alpha }_D=0$. If $i=n$ and $\alpha$ has zero image at every point of $K$, restriction of ${\alpha }_D$ to each chosen ball is zero because its center lies in $K$ and point restriction is injective there by step 2.1. Hence ${\alpha }_D$ is zero at every point of $D$, and pointwise injectivity in step 3.1 makes it zero. Finally, choose finitely many closed coordinate balls contained in the chart whose interiors cover $K$. Their union has its canonical class by step 3.1; restricting it realizes all $u_x$ on $K$. This proves existence, uniqueness, and vanishing for arbitrary compact coordinate supports.

Finite chart gluing proves the boundaryless statement for every compact support. [F2, F4, step 1.2, step 4.1] Choose finitely many coordinate balls whose smaller closed balls cover $K$ and whose closures lie in larger coordinate charts; compactness supplies the finite family. Put $K_j=K\cap {\bar{B}}_j$. Each $K_j$ satisfies step 4.1. When adjoining $K_j$ to the preceding union, the intersection is a finite union of compact sets contained in the larger chart for $K_j$, hence is one compact coordinate support and also satisfies step 4.1. Induction using the support sequence of step 1.2 gives vanishing above $n$, degree-$n$ pointwise injectivity, and the unique class with local values $u_x$ on all of $K$. If $K\subseteq L$, restriction of $[M{]}_L^{\mathrm{t}\mathrm{w}}$ has those same values on $K$, so uniqueness gives $[M{]}_K^{\mathrm{t}\mathrm{w}}$.

Collar cores construct the unique relative twisted class for a compact manifold with boundary. [F2, F3, F6, F7, step 5.1] Let $A=\mathrm{\partial }M$ and $N=M\setminus A$. For sufficiently small $d>0$, let $C_d$ be the collar of height below $d$ and put $K_d=M\setminus C_d\subset N$. Excision and the coefficient identification in [F6] give $H_n(N,N\setminus K_d;{\mathcal{O}}_N^R)\cong H_n(M,C_d;{\mathcal{O}}_M^R)$. Since $C_d$ retracts onto $A$, natural pair sequences and the five-lemma comparison in [F7] make $H_n(M,A;{\mathcal{O}}_M^R)\to H_n(M,C_d;{\mathcal{O}}_M^R)$ an isomorphism. Define $[M,A{]}^{\mathrm{t}\mathrm{w}}$ as the inverse image of $[N{]}_{K_d}^{\mathrm{t}\mathrm{w}}$. Nested collar cores and support compatibility in step 5.1 show that the inverse image is independent of $d$. Every interior point lies in some $K_d$, so this class has all prescribed local images. If two relative classes did, their difference maps for any core to a class with zero point values, which is zero by step 5.1; the displayed isomorphism then makes the difference zero.

The pair boundary has exactly the outward-normal-first boundary class. [F2, F3, F6, step 1.1, step 5.1, step 6.1] In a collar half-ball, take a local boundary orientation cycle and cross it with the collar interval oriented from positive height toward the boundary, placing this outward direction first. Give the product chain the corresponding ambient orientation-system coefficient from [F6]. Its relative boundary at the terminal boundary face is the boundary orientation cycle; all remaining faces lie off the core or cancel in pairs. Thus the pair connector sends the local value of $[M,A{]}^{\mathrm{t}\mathrm{w}}$ to $u_x$ at each boundary point. Connector naturality and excision in [F2]--[F3] globalize the calculation. Since $A$ is compact and boundaryless, pointwise uniqueness in step 5.1 gives $\mathrm{\partial }[M,A{]}^{\mathrm{t}\mathrm{w}}=[A{]}^{\mathrm{t}\mathrm{w}}$ with the stated sign.

Empty, zero-dimensional, disconnected, and choice cases are accounted for. [F2, F4, F5, step 1.1, step 1.2, step 4.1, step 5.1, step 6.1, step 7.1] When $A=\mathrm{\varnothing }$, step 6.1 is the support class with $K=M$. Empty manifolds, empty supports, and the zero ring give zero groups and their unique classes; compact zero-manifolds have empty boundary. Components of a manifold are open because connected coordinate balls lie in the component of each point by [F4]--[F5]. Their open cover of a compact support has a finite subcover, so only finitely many components meet it; the intrinsic finite-chain construction splits over these components. All singular generators, including degenerate simplices, remain in [F2]. Every cover and ball selection above is reduced by compactness to one finite list, and no global orientation, path family, component basepoint family, or infinite family of primitives is selected. Hence no AC is used. No biconditional is asserted. ∎