Singular Cochains Mayer Vietoris and Smooth Singular Comparison

Define $f_{\mathrm{\#}}(\sum a_{\sigma }[\sigma ])=\sum a_{\sigma }[f\circ \sigma ]$. Composites are continuous, the sum is finite, and collecting equal images preserves addition and real scalar multiplication. Under the real tensor identification this is exactly the map in [F2]. In negative degrees it is the unique map between zero spaces.

For $k\ge 1$, $\mathrm{\partial }{f}_{\mathrm{\#}}[\sigma ]={\sum }_{i=0}^k(-1{)}^i[f\circ \sigma \circ {\delta }_i]=f_{\mathrm{\#}}\mathrm{\partial }[\sigma ]$. Real linearity extends equality to all chains; for $k\le 0$ both composites are zero. This includes constant and degenerate simplices because the computation does not discard any face.

On every generator, $g_{\mathrm{\#}}{f}_{\mathrm{\#}}[\sigma ]=[g\circ f\circ \sigma ]=(g\circ f{)}_{\mathrm{\#}}[\sigma ]$ and $({\mathrm{id}}_X{)}_{\mathrm{\#}}[\sigma ]=[\sigma ]$. Finite linear extension proves both laws. They also hold on zero groups, including all chains of the empty space. On a point each nonnegative chain map induced by its identity is the identity of $\mathbb{R}$. All maps are given by formulas on supplied generators; no AC is used.

For $a:S\to \mathbb{R}$ define $L_a(\sum r_s[s])=\sum r_{s}a(s)$ and $A_a(\sum n_s[s])=\sum n_{s}a(s)$. Unique finite coefficients make both sums well-defined; finite distributivity proves real linearity of $L_a$ and additivity of $A_a$. Scalar multiplication of additive maps is pointwise.

Evaluation sends $L_a$ and $A_a$ back to $a$ because their value on $[s]$ is $a(s)$. Conversely every real-linear $L$ satisfies $L(\sum r_s[s])=\sum r_{s}L([s])$, and every additive $A$ satisfies $A(\sum n_s[s])=\sum n_{s}A([s])$, including negative integers by additive inverses. Thus evaluation and extension are inverse in both cases and are real-linear. There is no finite-support condition on $a$.

Write $D[t]={\sum }_s{d}_{st}[s]$ with each column finite, and let $D_{\mathbb{R}}$ use the same formula over $\mathbb{R}$. Both $L_a{D}_{\mathbb{R}}$ and $A_{a}D$ have value ${\sum }_s{d}_{st}a(s)$ on $[t]$, so the identifications commute with these maps. A signed face sum is such a finite column, even when repeated faces combine. Zero columns and the degree-zero boundary give zero values.

If $S=\mathrm{\varnothing }$, all three spaces are zero, with the unique empty function. If $S$ is a singleton, evaluation is the usual identification with $\mathbb{R}$; in negative singular degrees the groups are zero by convention. All inverse maps have specified formulas on the supplied formal generators, so no representative or basis selection and no AC is used.

Set $f^{\ast }\phi =\phi f_{\mathrm{\#}}$. The chain equation [F1] gives $\delta f^{\ast }=f^{\ast }\delta$, hence $f^{\ast }$ preserves cocycles and coboundaries and induces a real-linear map on the quotient [F2]. The generator identity and composition laws in [F1] become ${\mathrm{id}}^{\ast }=\mathrm{id}$ and $(gf{)}^{\ast }=f^{\ast }{g}^{\ast }$.

Put $K^k\phi =\phi P_{k-1}$ for $k\ge 1$ and $K^k=0$ for $k\le 0$. For a degree-$k$ cochain and $c\in C_k(X;\mathbb{R})$, [F3] gives $(g^{\ast }-f^{\ast })\phi (c)=\phi (\mathrm{\partial }{P}_{k}c+P_{k-1}\mathrm{\partial }c)=(K^{k+1}\delta \phi +\delta K^k\phi )(c)$. In degree zero the second summand is zero, exactly as in [F3]; negative degrees are zero.

If $\phi$ is a cocycle, step 1.2 says $g^{\ast }\phi -f^{\ast }\phi =\delta K^k\phi$, so their classes coincide. If $f:X\to Y$ has a supplied homotopy inverse $h:Y\to X$, apply this equality to $hf\simeq {\mathrm{id}}_X$ and $fh\simeq {\mathrm{id}}_Y$; step 1.1 yields $f^{\ast }{h}^{\ast }=\mathrm{id}$ and $h^{\ast }{f}^{\ast }=\mathrm{id}$. Thus $f^{\ast }$ is an isomorphism.

Empty spaces and negative groups have only zero maps, and on a point the identity acts identically on $H^0=\mathbb{R}$. Constant homotopies and degenerate simplices need no normalization: [F3] applies to their full signed prisms as well. Only the given homotopy and its explicit finite prism are used, so no AC is needed.

Regard chains in $U$, $V$, and $U\cap V$ as chains in $X$ by their basis inclusions: a map with image in a subspace has a unique corestriction, continuous by the subspace topology. A face of a small simplex remains small. Hence these spaces and the span $C^{U,V}$ are subcomplexes, and both displayed arrows commute with boundary. The sign choice equals [F2] precomposed by $-\mathrm{id}$ on the overlap.

The map $i$ is injective since its second component is $c$. To split a finite small chain, assign a simplex to the $U$ component whenever its image lies in $U$, and otherwise to $V$. This formula gives a preimage under $j$, proving surjectivity without choosing from an arbitrary family.

If $a+b=0$, coefficient comparison in the supplied $X$ basis forces coefficients of $a$ outside the overlap to vanish, and likewise for $b$. Thus $b=-a$ is an overlap chain, and $(a,b)=i(b)$. Conversely $ji(c)=-c+c=0$. These prove equality of kernel and image. If an open set or overlap is empty the same argument gives the appropriate zero term; when $U=V=X$ it is the diagonal signed sequence. In negative degrees all terms are zero, and degree zero uses the same basis argument with zero differential. Constant simplices remain generators, and no AC is used.

For each $U$-simplex $\sigma$, set $(E_A^U\eta )([\sigma ])=\eta ([\sigma :A])$ if $\sigma ({\mathrm{\Delta }}^k)\subseteq A$, and set it to zero otherwise; $\sigma :A$ is the unique continuous corestriction. Extend by the finite sum formula of [F1]. Membership in $A$ and the unique corestriction specify a function, so no selection of a vector-space basis or representatives occurs. This formula is linear in $\eta$.

On an $A$-simplex the first clause always applies, so restriction of $E_A^U\eta$ is $\eta$ on each generator and hence on every chain. If $A=\mathrm{\varnothing }$ this is the zero map; if $A=U$ it is the identity. At degree zero it extends point functions by zero, including a one-point subspace. In negative degrees define the unique zero section.

To check that this is not generally a cochain map, take $U=[0,1]$, $A=\{0\}$, and the zero-degree cochain $\eta (0)=1$. On the identity path $\sigma (t)=t$, $\delta E_A^U\eta ([\sigma ])=0-1=-1$. But $\delta \eta =0$, since every path in $A$ is constant and has equal endpoints, so $E_A^U\delta \eta ([\sigma ])=0$. Thus the two composites differ. Constant and repeated simplices are fully allowed in the formulas. All degreewise section assertions, including the zero cases, follow from the prescribed basis values and require no AC.

Since $i,j$ commute with boundary, precomposition by them commutes with coboundary. Their dual maps are exactly $a$ and $b$ with the displayed signs. Also ${\delta }^2\lambda =\lambda {\mathrm{\partial }}^2=0$ on small chains, so the first term is a cochain complex, including zero groups in negative degrees.

If both restrictions of $\lambda$ vanish then $\lambda$ vanishes on every small generator, so $a$ is injective. A pair $(\phi ,\psi )$ is in $\ker b$ precisely when the functions agree on overlap simplices. Define $\lambda$ on a small simplex to be $\phi$ on $U$ simplices and $\psi$ on the others. Agreement makes its restrictions the given pair. Extending by finite real sums gives a unique functional. Conversely restrictions of a single functional agree on the overlap, proving $\ker b=\mathrm{im}a$.

For $\eta \in C^k(U\cap V;\mathbb{R})$, $b(-E_{U\cap V}^U\eta ,0)=\eta$ by [F2], proving degreewise surjectivity with the required minus sign. This lift is not used as a cochain map. Empty opens or overlap give zero terms and the same formulas; if $U=V=X$, agreeing pairs are $(\lambda ,\lambda )$ and $b(\phi ,\psi )=\psi -\phi$. This includes a one-point space. The argument applies in degree zero, on repeated simplices, and on the zero groups in negative degrees, without any AC.

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.

Postcomposition by $f$ takes an $X$-simplex in $U$ to a $Y$-simplex in $U^{\mathrm{\prime }}$, and likewise for $V$ and the intersections. It therefore induces chain maps on all three terms of the small-chain sequence. Composition commutes with face restriction, the signed overlap inclusion and sum. Dualizing gives cochain maps $D_Y\to D_X$, $E_Y\to E_X$, $F_Y\to F_X$ commuting with the two arrows in [F1].

By [F2] this cochain diagram commutes with the connectors into $H(D)$. Explicitly, if $be=c$ and $\delta e=a(d)$ in the $Y$ row, the images satisfy the same equations in the $X$ row, so the image of $d$ is the connecting representative of the image of $c$. This uses an image of a supplied lift; it does not claim that zero extension is natural.

Write $I_X,I_Y$ for the small-chain inclusions and $f_{\mathrm{s}\mathrm{m},\mathrm{\#}}$ for the small-chain map. The equation $f_{\mathrm{\#}}{I}_X=I_Y{f}_{\mathrm{s}\mathrm{m},\mathrm{\#}}$ holds on every generator. Hence the canonical isomorphisms $\theta =H(I^{\ast })$ of [F1] commute with pullback. Transporting step 2.1 by their inverses proves the displayed connector identity. The restriction and difference squares already commute by step 1.1, giving the full sequence claim.

Empty intersections yield zero connector domains; empty spaces yield zero groups. In degree zero the same lift calculation applies, with negative primitives zero. On one-point spaces or identical covers the connector is zero by [F1]. Constant or degenerate simplices still satisfy the generator equation in step 3.1. All maps are supplied by postcomposition; neither natural choices of chain inverses nor AC are required.

Composition is smooth also in the boundary case: around a point choose charts for the two maps, take local Euclidean smooth extensions of their coordinate representatives from [F3], and shrink the first extension domain so its image is inside the domain of the second. Their ordinary smooth composite extends the coordinate representative of the composite on the original half-space domain. The same coordinate argument for identity uses the Euclidean identity. This is the chart argument of [F2], with the local extension requirement explicitly respected.

If $\bar{\sigma }:O\to M$ extends a smooth simplex, $f\bar{\sigma }:O\to N$ is smooth by step 1.1 and takes values in $N$. Thus $f_{\mathrm{\#}}[\sigma ]=[f\sigma ]$ preserves smooth generators and has a unique finite real-linear extension. For $k\ge 1$, $\mathrm{\partial }{f}_{\mathrm{\#}}[\sigma ]={\sum }_i(-1{)}^i[f\sigma {\delta }_i]=f_{\mathrm{\#}}\mathrm{\partial }[\sigma ]$; for $k\le 0$ both sides are zero.

Set $f^{\ast }\phi =\phi f_{\mathrm{\#}}$. Then $\delta f^{\ast }\phi =\phi f_{\mathrm{\#}}\mathrm{\partial }=\phi \mathrm{\partial }{f}_{\mathrm{\#}}=f^{\ast }\delta \phi$, so cycles and boundaries are preserved and a quotient pullback is defined. On each smooth generator $(gf{)}_{\mathrm{\#}}=g_{\mathrm{\#}}{f}_{\mathrm{\#}}$ and $(\mathrm{id}{)}_{\mathrm{\#}}=\mathrm{id}$ by composition of maps; precomposition reverses these laws, and passing to quotient classes preserves them.

On empty manifolds or negative degrees the maps are the unique zero maps. At degree zero they act by the point maps; on a one-point identity they are the identity of $\mathbb{R}$. Degenerate simplices and constant maps, including those landing in the boundary, retain their target-valued extensions through step 2.1. Each local extension is used for one simplex only; no simultaneous choice and no AC is used.

If $a:{\mathrm{\Delta }}^j\to {\mathrm{\Delta }}^k$ is affine, it extends to an affine map $\bar{a}:A^j\to A^k$. The open inverse image ${\bar{a}}^{-1}(O)$ contains ${\mathrm{\Delta }}^j$, and $\bar{\sigma }\bar{a}$ extends $\sigma a$ smoothly into $M$. Each simplex in $S\sigma$ and $T\sigma$ has this form: coning affine simplices appends a fixed barycenter vertex and so remains affine, at every stage of the finite recursion [F2]. Therefore both operators preserve smooth chains. Their affine images stay in ${\mathrm{\Delta }}^k$, so they also preserve any specified image-containing subset of $M$.

Let $B$ be a smooth target-valued extension of $(x,t)\mapsto H(\sigma (x),t)$ to an open neighbourhood $W$ of ${\mathrm{\Delta }}^k\times [0,1]$. Each prism simplex ${\lambda }_i:{\mathrm{\Delta }}^{k+1}\to {\mathrm{\Delta }}^k\times [0,1]$ is affine. Its affine extension has open inverse image of $W$ containing ${\mathrm{\Delta }}^{k+1}$, and $B{\lambda }_i$ is the required extension into the target. Thus every term of the signed prism is smooth. The equality [F3] is an equality in this subcomplex because every term is now in it.

For a smooth homotopy $H:M\times [0,1]\to N$, with smoothness interpreted by local coordinate extensions also at the time endpoints, replace it by $\widehat{H}(p,t)=H(p,s(t))$ for all real $t$. This is smooth: near an endpoint use a local coordinate extension of $H$ and compose with $(p,t)\mapsto (p,s(t))$; near an interior time ordinary smooth composition suffices. It is target-valued for every real $t$ because $s(t)\in [0,1]$. Then $(x,t)\mapsto \widehat{H}(\bar{\sigma }(x),t)$ is smooth on $O\times \mathbb{R}$ and satisfies step 2.1. Its endpoint maps are exactly those of $H$, so [F3] gives the same difference of induced chain maps.

The qualification in step 2.1 is necessary for an unmodified prism with a boundary target. Take $M$ a point and $H(t)=t$ into $N=[0,\mathrm{\infty })$. This is a smooth homotopy on the interval, but its prism is the path $t\mapsto t$, which has no smooth target-valued extension across $0$: any such nonnegative extension has a local minimum at $0$ and derivative zero, whereas its right derivative would be one. Time flattening avoids this obstruction. In degree zero $S=1$ and $T=0$; the flattened homotopy prism is still a smooth path. Empty chains and empty domains give zero operators, repeated affine vertices are allowed, and all formulas are finite and choice-free.

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.

If $n=0$, take $O=\mathrm{\varnothing }$ and the empty map. Otherwise the face maps glue continuously to $g:\mathrm{\partial }D\to N$, because the boundary is a finite closed union and the maps agree on all intersections. Fix the data in [F2]. The only infinite-choice use in this proof is the countable embedding construction inherited there.

At $p\in \mathrm{\partial }D$ write $J=\{i:{\lambda }_i(p)=0\}$ for its incident faces, where ${\lambda }_i$ are barycentric coordinates. Choose one $k\mathrm{\notin }J$ and use the other $n$ coordinates as affine coordinates on $E$. Shrink around $p$ so every nonincident ${\lambda }_i$ remains positive. For a nonempty $I\subseteq J$, define $P_I$ by setting the coordinates in $I$ to zero and adding their sum to ${\lambda }_k$. Then $P_I{P}_L=P_{I\cup L}$, $P_{I}p=p$, and $P_I(D)\subseteq D$ by nonnegativity of the retained and added coordinates.

On $E_I=\{{\lambda }_i=0:i\in I\}$ take a smooth Euclidean extension $a_I$ of $jg$ on its actual face intersection near $p$: restrict an extension of $j{g}_i$ for one $i\in I$ to $E_I$. The intersections of the finitely many inverse extension domains give an open $V_p$ where every $a_I{P}_I$ is defined. Set $A_p={\sum }_{\mathrm{\varnothing }\mathrm{\ne }I\subseteq J}(-1{)}^{\mathrm{\mid }I\mathrm{\mid }+1}{a}_I{P}_I$. For $x\in V_p\cap \mathrm{\partial }D$, choose $j_0\in J$ with ${\lambda }_{j_0}(x)=0$. Pair each nonempty $I$ omitting $j_0$ with $I\cup \{j_0\}$. Their projected points coincide and lie in the actual intersection face, by step 2.1, so compatibility gives equal values with opposite signs. Only $I=\{j_0\}$ remains, giving $A_p(x)=jg(x)$. No agreement of the arbitrary extensions outside the actual simplex intersections was assumed.

Consider all local data from steps 2.1–3.1 together with balls $B(p,r)$ whose closed doubled balls lie in $V_p$. They form a cover of compact $\mathrm{\partial }D$. Extract finitely many, indexed by $a$, with extensions $A_a$ on neighbourhoods of their closed doubled balls. Define ${\eta }_a(x)=1-s((\mathrm{\parallel }x-p_a{\mathrm{\parallel }}^2-r_a^2)\mathrm{/}(3r_a^2))$. It equals one on the radius-$r_a$ ball and zero outside the radius-$2r_a$ ball. The product ${\eta }_a{A}_a$, extended by zero, is globally smooth: its support is contained in a closed ball strictly inside the domain of $A_a$. This uses a finite subcover of all eligible data, not a simultaneous point-indexed choice.