Smooth singular mayer vietoris sequence

For a smooth simplex $\sigma$, let $a(\sigma )$ be the least nonnegative integer such that $S^{a(\sigma )}\sigma$ is small; [F5] supplies existence. Set $m=0$ on vertices and recursively $m(\sigma )=\max (\{a(\sigma )\}\cup \{m(\sigma {\delta }_j){\}}_j)$. Each maximum is finite, faces have smaller dimension, and [F3] ensures all face and subdivision chains remain smooth. A small simplex has $m=0$, by induction on its faces.

Define $D_q={\sum }_{i=0}^{q-1}T{S}^i$, so $1-S^q=\mathrm{\partial }{D}_q+D_q\mathrm{\partial }$ by telescoping [F4]. Set $D\sigma =D_{m(\sigma )}\sigma$ and $R=1-\mathrm{\partial }D-D\mathrm{\partial }$, extending on the supplied smooth basis. Then $\mathrm{\partial }R=R\mathrm{\partial }$. On a simplex, $R\sigma =S^{m(\sigma )}\sigma +D_{m(\sigma )}\mathrm{\partial }\sigma -D\mathrm{\partial }\sigma$. For each face $\tau$, its correction is the signed sum ${\sum }_{i=m(\tau )}^{m(\sigma )-1}T{S}^i\tau$; every term is smooth and small because $S^{m(\tau )}\tau$ is small and $S,T$ preserve that property. Thus $R$ lands in $A$.

With $I:A\to C$ and $r=R:C\to A$, the identities are $1-Ir=\mathrm{\partial }D+D\mathrm{\partial }$ and $rI=1$, since $D$ vanishes on small simplices and their faces. Dualizing these actual equations gives ${\theta }_{\mathrm{\infty }}=H(I^{\ast }):H_{\mathrm{\infty }}^k(M)\cong H^k({\mathrm{Hom}}_{\mathbb{R}}(A,\mathbb{R}))$: the cochain homotopy is precomposition with $D_{k-1}$. This proves the needed equivalence within smooth chains, independently of any smooth/continuous comparison theorem.

The smooth bases for $U$ and $V$, viewed in $M$, have intersection precisely the smooth overlap basis. To check the corestriction clause, restrict any extension of an overlap-valued simplex to the open inverse image of $U\cap V$, which contains its simplex. Thus the argument [F1] applies to these bases: restrictions inject the small dual into the pair of cochains, agreeing pairs glue uniquely by priority-U values, and $(-E_U\eta ,0)$ maps to $\eta$ under $V-U$ difference. Signed face restrictions make both arrows cochain maps. Apply [F6] to this explicitly exact smooth row and identify its first cohomology through ${\theta }_{\mathrm{\infty }}$ from step 3.1. This proves the sequence and initial injection.

For clarity, lift an overlap cocycle $c$ to a pair $e$; its differential equals $a(d)$ for a unique small cocycle $d$. The connector is ${\theta }_{\mathrm{\infty }}^{-1}[d]$. Changing the lift by $a(t)$ changes $d$ by $\delta t$; changing $c$ by $\delta c_0$ and adding the differential of a lift of $c_0$ leaves $d$ unchanged. For a smooth ordered-cover map, postcomposition preserves each small smooth basis and commutes with inclusions and signed arrows. The image of $e$ is a lift of the image of $c$, proving connector naturality, and the natural inclusion square transports it through ${\theta }_{\mathrm{\infty }}$.

Negative degrees have zero terms; at degree zero all vertices are small, $m=D=0$, and no negative primitive exists. Empty opens and overlap give zero terms; when $U=V=M$ the row is diagonal then difference, with connector zero by the lift $(0,c)$. This includes one-point and empty manifolds. Degenerate simplices have the same face recursion; no quotient discards them. Least integers, finite maxima and prescribed zero values give all constructions without AC, including boundary targets.