Top homology of a connected manifold

The chain coefficient map ${\iota }_R:C_{\ast }(-;\mathbb{Z})\to C_{\ast }(-;R)$ sends $\sum m_j{\sigma }_j$ to $\sum (m_j{1}_R){\sigma }_j$. It commutes directly with boundary, passage to relative quotients, and maps of pairs. We now verify on the stated interfaces that its map on an integral local stalk takes a generator to an $R$-module generator, consistently with the abstract sphere and local-group computations in [F5] and [F6]. For $n\ge 1$, choose a chart carrying $x$ to $0$ in an open ball $B$, and a concentric closed disk $D$ with boundary sphere $S$. The radial formula $h(y,t)=(1-t+t\frac{\rho }{\parallel y\parallel })y$ deformation retracts $B\setminus \{0\}$ onto $S$, where $\rho$ is the radius of $D$: its norm is $(1-t)\parallel y\parallel +t\rho$, so it stays nonzero and inside $B$. The disk and $B$ are contractible, hence [F11] and the natural pair sequences show that inclusion $(D,S)\to (B,B\setminus \{0\})$ induces an isomorphism in degree $n$; for $n=1$ the connecting terms are the augmentation kernels in reduced $H_0$, exactly as in [F9]. By [F10], the chart map and excision then identify this ball-pair group with $H_n(M,M\setminus \{x\};G)$ for $G=\mathbb{Z}$ and for $G=R$. The sphere calculation [F5] gives a primitive positive boundary generator with coefficient $1$, and [F9] gives its unique preimage in $H_n(D,S;\mathbb{Z})$. The coefficient map sends the boundary generator to the same alternating cycle with coefficient $1_R$ (for $n=1$, it sends $(1,-1)$ to $(1_R,-1_R)$ in the augmentation kernel). Naturality of [F9] therefore sends its relative preimage to the preimage of the coefficient-$1_R$ boundary generator. Under the chart and excision isomorphisms this is an $R$-module generator of the local stalk, also when $R$ is the zero ring. Thus the induced integral local generator maps to an $R$-generator. For $n=0$, the same conclusion follows directly from the point class with coefficient $1$. Finally, the transports in [F2] and [F3] are composites of the same ball restriction isomorphisms, so their naturality with ${\iota }_R$ is the chain-level commutation just established. An integral stalk automorphism is multiplication by $+1$ or $-1$, since its value on a generator must again be a generator. Hence the corresponding loop transports on an $R$ stalk have those same signs.

Any two points of $M$ can be joined by a path. Indeed, the subset of points reachable from one fixed point is open, since paths can be extended within convex chart neighborhoods. Every other path-reachability class is open for the same reason, so its complement is open. Connectedness and nonemptiness force the reachable subset to be all of $M$. In a ball trivialization, a continuous local-system section is constant along a path segment: its discrete coordinate is continuous on a connected interval by [F7], and a nonconstant value would separate that interval by a fiber and its complement. Therefore a section is determined everywhere by its value at any one point, using finite path subdivisions as in [F2].

For any finite singular cycle $z$ in $M$, its image is compact: the finitely many simplex domains are compact by [F8], continuous images are compact by pulling back covers, and finite unions preserve compactness. Cover this image by interiors of closed coordinate balls lying inside larger charts, and take a finite subcover. The union $U$ of the smaller open balls contains the image of $z$ and has compact closure, since its closure lies in the finite union of the closed balls and is closed there. Put $V=M\setminus \bar{U}$, and $W=U\cup V$. These are open, with $U,V$ disjoint. Both $M\setminus V=\bar{U}$ and $M\setminus W=\bar{U}\setminus U$ are compact.

Fix a base point $x_0$ and one integral generator there, whose coefficient extension identifies the $R$ stalk with $R$ by step 1.1. A section's value $r$ is fixed by all loop transports. Conversely, for any such fixed $r$, define its value at $y$ by transport along any path from $x_0$ to $y$. Paths exist by step 1.2. Two choices give the same value because traversing one and reversing the other is a loop fixing $r$. Thus there is a unique prescribed value at each point, defining a function without choosing a path simultaneously for every point. On a ball its values are the local restrictions of one class, so the resulting section is continuous. This gives an $R$-linear bijection between sections and the common fixed submodule of the loop signs. It is $R$ if all signs are positive, and $R[2]$ if a negative sign occurs.

For the triple $V\subset W\subset M$, the quotient sequence $0⟶C_{\ast }(W;R)\mathrm{/}{C}_{\ast }(V;R)⟶C_{\ast }(M;R)\mathrm{/}{C}_{\ast }(V;R)⟶C_{\ast }(M;R)\mathrm{/}{C}_{\ast }(W;R)⟶0$ is degreewise exact: the first map is inclusion modulo $C_{\ast }(V)$, and the kernel of the last quotient consists exactly of the classes represented in $C_{\ast }(W)$. By [F4] it gives $H_{i+1}(M,W;R)⟶H_i(W,V;R)⟶H_i(M,V;R)\mathrm{.}$ Every singular simplex in $W$ lies in $U$ or $V$, since the inverse images of these disjoint open sets would otherwise separate its connected domain ([F7]). Thus $C_{\ast }(W)\mathrm{/}{C}_{\ast }(V)=C_{\ast }(U)$, compatibly with maps of the cycle $z$. By [F1] and the compactness in step 1.3, the left group is zero for $i\ge n$. Therefore $H_i(U;R)\to H_i(M,V;R)$ is injective for $i\ge n$. For $i>n$, its target also vanishes by [F1], so $z$ bounds already in $U$. This proves $H_i(M;R)=0$ for all $i>n$, whether or not $M$ is compact.

A section is an orientation exactly when its base-point value is a unit: all transports are module isomorphisms and [F3] characterizes generators as unit multiples of one generator. If the fixed submodule is $R$, it contains $1$ and $M$ is $R$-orientable. If it is $R[2]$ and contains a unit $u$, then $2u=0$ implies $2=0$ after multiplication by $u^{-1}$, so $R[2]=R$ and again $M$ is $R$-orientable. Consequently the fixed submodule is $R$ in the $R$-orientable case, and exactly $R[2]$ in the non-$R$-orientable case. No claim that the latter subgroup vanishes is justified without a restriction on $R$.

For compact $M$, any class $a\in H_n(M;R)$ yields a continuous section of point restrictions: over a coordinate ball, first restrict $a$ to its ball-supported group, and its subsequent point values form exactly a basic section of [F3]. By [F1] with support $M$, this map from classes to sections is bijective and all higher groups vanish. Evaluation at $x_0$ and steps 2.1–3.1 give the stated injective stalk map and its image. In the oriented case the section of generators corresponds to the fundamental class of [F3], and multiplying it by $r$ gives the class with local coefficient $r$.

Suppose $M$ is noncompact and $z$ is a degree-$n$ cycle. Its point restrictions define a continuous section just as in step 4.1. Outside its compact image the restriction is zero because the chain is in the omitted subspace. Such a point exists by noncompactness, and step 1.2 forces the section to be zero everywhere. By [F1], the image of $z$ in $H_n(M,V;R)$, supported on compact $\bar{U}$, is therefore zero. The injection of step 2.2 makes $[z]=0$ in $H_n(U;R)$ and hence in $H_n(M;R)$. This uses the triple injection, not just vanishing of all point restrictions.

A connected nonempty zero-manifold is one point, so is compact and has $H_0=R$, consistent with step 4.1. The zero ring has the unique orientation and every displayed module is zero; a nonorientable case does not arise for it. Characteristic two makes both signs act as identity. The two signs in step 1.1 include all monodromy possibilities, without a hidden assumption of integral orientability. Every global construction used unique values or a finite subcover; the base point and its generator are individual witnesses, not a family selected by AC. All chain calculations use unnormalized simplices.