Mayer vietoris sequence in real singular cohomology

Let $r$ be the inverse chain map supplied by [F2], with $rI=1$ and $1-Ir=\mathrm{\partial }T+T\mathrm{\partial }$. Precomposition gives $I^{\ast }:C^{\bullet }(X)\to D$ and $r^{\ast }:D\to C^{\bullet }(X)$. Then $I^{\ast }{r}^{\ast }=1$ and $1-r^{\ast }{I}^{\ast }=\delta K+K\delta$, where $K^k\phi =\phi T_{k-1}$ and $K^k=0$ for $k\le 0$. Indeed evaluation on any degree-$k$ chain turns this equality into the displayed chain identity. Consequently $\theta =H(I^{\ast })$ is an isomorphism with inverse $H(r^{\ast })$.

For a cocycle $c\in F^k$, take a lift $e\in E^k$ with $be=c$. Then $b\delta e=0$, so $\delta e=a(d)$ for a unique $d\in D^{k+1}$. Since $a$ is injective, $a\delta d={\delta }^{2}e=0$ implies $\delta d=0$. Changing $e$ by $a(t)$ changes $d$ by $\delta t$; changing $c$ by $\delta c_0$ and lifting $c_0$ to $e_0$ allows replacement of $e$ by $e+\delta e_0$, leaving $d$ unchanged. Thus $\mathrm{\Delta }[c]={\theta }^{-1}[d]$ is well-defined and linear, because lifts add and scale. This is the lift-differential convention of [F3] and [F4].

Apply [F3] to [F1]. The finite direct sum has cohomology $H(E)=H(U)\oplus H(V)$: cycles and boundaries are componentwise, with just two primitives for a boundary pair. Replace $H(D)$ by $H(X)$ via $\theta$. Since $a{I}^{\ast }$ restricts a full cochain to $U$ and $V$, the first map is the actual $\alpha$; the second is the displayed $\beta$, and the connector is step 2.1. This proves exactness throughout. At degree zero the preceding groups are zero, giving initial injectivity.

For the sign comparison, the AT row and this row have the same $D,E,a$, whereas $b=-b_{\mathrm{A}\mathrm{T}}$. If $e$ lifts $c$ in the AT row, then $-e$ lifts $c$ in this row and its differential is $-a(d)$. Therefore $\mathrm{\Delta }=-{\mathrm{\Delta }}_{\mathrm{A}\mathrm{T}}$ under the same $\theta$. Equivalently the row comparison is identity on $D,E$ and minus identity on $F$.

If an open set is empty, restriction to the other is an identity and the overlap terms vanish. If $U=V=X$, $\alpha$ is diagonal, $\beta (u,v)=v-u$, and the cochain lift $c\mapsto (0,c)$ makes the connector zero. This covers one-point and empty spaces. Degree zero and all negative degrees were handled in steps 1.1 and 3.1; degenerate simplices are included in the supplied chain equivalence. The inverse uses least subdivision depths in [F2], and the lifts in [F1] are canonical zero extensions, so no AC is introduced.