Generalized Cohomology and the Atiyah Hirzebruch Spectral Sequence

First record two consequences of the pair axioms, for use in construction (B). The sequence of $(Z,Z)$ gives $h^n(Z,Z)=0$: the adjacent maps $h^n(Z)\to h^n(Z)$ are identities. For a based complex $Z$, the retraction $Z\to \ast$ splits $h^n(Z)\to h^n(\ast )$. Thus the sequence of $(Z,\ast )$ identifies $h^n(Z,\ast )$ naturally with its kernel and gives $h^n(Z)=h^n(Z,\ast )\oplus h^n(\ast )$. For a based subcomplex $D\subseteq Z$, remove the split point summands from the pair sequence to obtain the exact sequence $\cdots \to h^n(Z,D)\to h^n(Z,\ast )\to h^n(D,\ast )\to h^{n+1}(Z,D)\to \cdots$. Indeed the pair boundary kills the point summand because it is in the image of $h^n(Z)$.

The quotient axiom for a pair theory follows from the stated axioms, rather than being an additional assumption. For nonempty $D\subseteq Z$, form $T=Z{\cup }_{D}CD$, where $CD$ here is the ordinary cone with its tip as basepoint; equivalently it is the reduced cone on $D_{+}$. The collapse $T\to Z\mathrm{/}D$ is a homotopy equivalence by [F2] and [F3]. Its restriction $CD\to \ast$ is also a homotopy equivalence. The natural pair sequences and [F6] therefore make $h^n(Z\mathrm{/}D,\ast )\to h^n(T,CD)$ an isomorphism. Excision for the subcomplex cover $T=Z\cup CD$ makes $h^n(T,CD)\to h^n(Z,D)$ an isomorphism. Their composite is precisely the pullback of the quotient map of pairs $(Z,D)\to (Z\mathrm{/}D,\ast )$, so this identification is natural. For $D=\mathrm{\varnothing }$, additivity identifies $h^n(Z_{+},\ast )$ with $h^n(Z,\mathrm{\varnothing })$. This proves the same assertion with the stipulated convention $Z\mathrm{/}\mathrm{\varnothing }=Z_{+}$.

Starting with a reduced theory, define $h^n(X,A)={\tilde{h}}^n(C_{A_{+}\to X_{+}})$. Maps and homotopies of pairs induce maps and homotopies of these cofibers, proving functoriality and (H). Cofibration collapse identifies the cofiber with $X\mathrm{/}A$ when $A\mathrm{\ne }\mathrm{\varnothing }$. For $A=\mathrm{\varnothing }$, $A_{+}$ is a point, and its reduced cone adds nothing to $X_{+}$, so the cofiber is $X_{+}$. In all cases the cofiber exact sequence gives (LES), with boundary $q_i^{\ast }\sigma$.

For subcomplexes $X=U\cup V$, the map $U\mathrm{/}(U\cap V)\to X\mathrm{/}V$ is a based homeomorphism under the empty-quotient convention. If $U\cap V$ is empty and $V$ is nonempty, the collapsed $V$ is the adjoined isolated basepoint of $U_{+}$; if $V$ is empty both quotients are $X_{+}$. In the remaining case both quotients have the same cell characteristic maps and weak topology. This proves (Ex). Also $(\coprod X_{\alpha }{)}_{+}=\bigvee (X_{\alpha }{)}_{+}$, including an empty family, and the cofiber of the wedge of these inclusions is the wedge of their cofibers, by their explicit quotient constructions. The reduced wedge axiom proves (Add).

Conversely set ${\tilde{h}}^n(Y)=h^n(Y,\ast )$. This is a homotopy-invariant functor and is zero at a point by step 1.1. For the reduced cone pair $(CY,Y)$, step 1.1 and based contractibility of $CY$ give an isomorphism $\mathrm{\partial }:h^n(Y,\ast )\to h^{n+1}(CY,Y)$. By step 1.2 the quotient map induces an isomorphism ${\kappa }_Y:h^{n+1}(\mathrm{\Sigma }Y,\ast )\to h^{n+1}(CY,Y)$. Define $\sigma ={\rho }_Y^{\ast }{\kappa }_Y^{-1}\mathrm{\partial }$ as in (B). All these maps are natural, so this is a natural suspension isomorphism.

For the wedge axiom, form a CW complex $Z$ from the disjoint union of the $Y_{\alpha }$ by adjoining one new vertex and an interval from it to each supplied basepoint. Let $T$ be the union of those intervals and their endpoints. The simultaneous linear contraction of the intervals to the new vertex contracts $T$; it is continuous in the CW weak topology. The quotient $Z\mathrm{/}T$ is the wedge with its CW topology. Excision for the subcomplex cover by $T$ and $\coprod Y_{\alpha }$, and then (Add), give $h^n(Z,T)\cong h^n(\coprod Y_{\alpha },\coprod \{{\ast }_{\alpha }\})\cong \prod h^n(Y_{\alpha },{\ast }_{\alpha })$. Step 1.2 identifies the left side with $h^n(\bigvee Y_{\alpha },\ast )$. The comparison is the map induced by summand inclusions, since all the quotient and excision maps restrict to those inclusions. For the empty family $Z=T=\ast$ and all groups are zero. Thus (W) holds.

To check the connector including its sign, use the reduced mapping cylinder $M_f=Y{\cup }_f(X\times I)$, attaching $X\times \{0\}$ to $Y$ and collapsing the basepoint track. Its free end $X\times \{1\}$ is a CW subcomplex, $M_f$ retracts onto $Y$, and $M_f\mathrm{/}(X\times \{1\})=C_f$ with the cone coordinate of [F2]. Define a map of pairs $(M_f,X\times \{1\})\to (CX,X\times \{0\})$ by sending $Y$ to the cone tip and $[x,t]$ to $[x,1-t]$. It is the identity on the identified copies of $X$. On quotients it induces ${\rho }_X{q}_f:C_f\to \mathrm{\Sigma }X$. Naturality of the pair boundary and the quotient isomorphisms of step 1.2 now identify the cylinder boundary on $h^n(X,\ast )$ with $({\rho }_X{q}_f{)}^{\ast }{\kappa }_X^{-1}\mathrm{\partial }=q_f^{\ast }\sigma$. The reduced pair sequence of step 1.1, together with the retraction, is exactly (E), with this connector. This proves both exactness and the normalization required by [F1].

Starting from a reduced theory, applying (A) and (B) yields $h^n(Y,\ast )={\tilde{h}}^n(C_{S^0\to Y_{+}})\cong {\tilde{h}}^n(Y_{+}\mathrm{/}{S}^0)={\tilde{h}}^n(Y)$; this is natural, not a claimed decomposition of $Y_{+}$. Check suspension as well. The cofiber of $Y\hookrightarrow CY$ is two reduced cones glued along their bases. Collapsing the attached cone gives the quotient model $CY\mathrm{/}Y=\mathrm{\Sigma }Y$, whereas the cofiber-to-suspension map collapses the first cone. These two maps differ, up to based homotopy, by reflection of the suspension coordinate: parametrize it by $s\in [0,2]$ from the first tip through the equator at $1$ to the second tip. If $\psi$ collapses the attached cone, then $\rho \psi$ is $[y,s]\mapsto [y,\min (s,1)]$, and $q$ is $[y,s]\mapsto [y,\max (s-1,0)]$. The maps $[y,s]\mapsto [y,\min (1,\max (0,s-a))]$ for $0\le a\le 1$ give the required based homotopy. Consequently the cone-pair boundary transported to $\mathrm{\Sigma }Y$ is ${\rho }_Y^{\ast }{\sigma }_{\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$. The additional reflection in (B) cancels this, since ${\rho }_Y^2=\mathrm{i}\mathrm{d}$. Thus the recovered suspension is the original one. All cofiber connectors then agree because both are $q_f^{\ast }\sigma$.

Starting from a pair theory, (B) followed by (A) gives $h^{\mathrm{\prime }n}(X,A)=h^n(C_i,\ast )$. Cofibration collapse and step 1.2 identify this naturally with $h^n(X,A)$, including $A=\mathrm{\varnothing }$. To check boundaries, replace $A_{+}\to X_{+}$ by its reduced mapping cylinder. The cylinder collapse is a map of pairs to $(X_{+},A_{+})$, an ordinary homotopy equivalence on total spaces and the identity on the identified copies of $A_{+}$. Naturality of the pair sequences and the five lemma make its relative pullback an isomorphism. Additivity identifies the latter relative and absolute maps with those for $(X,A)$. The coordinate calculation of step 3.1 identifies its connecting map under the quotient comparison with $q_i^{\ast }\sigma$. Hence the pair boundary recovered by (A) is the original one, with its sign. The same constructions commute with morphisms of theories, since they use only pullbacks, boundaries and inverses of natural isomorphisms.

The two natural comparisons of steps 4.1 and 4.2 preserve all structure maps and give the asserted inverse constructions. Empty spaces and empty subcomplexes use $X\mathrm{/}\mathrm{\varnothing }=X_{+}$, points have zero reduced groups, and no dimension axiom or coefficient restriction has entered. No family of arbitrary choices is used: cones, cylinders, quotients and the interval contraction are specified constructions on the supplied CW data.

All maps used are in the functors' domains. For $p\ge 1$ the skeleton of $S^p$ in each dimension $0\le k<p$ is its basepoint and in dimensions $k\ge p$ is the entire sphere. The same description holds for a finite wedge with its common vertex. Every based continuous map among these spaces therefore preserves every skeleton and is cellular. For $p=0$ every map of the discrete vertex sets is cellular. In particular this applies to $f$, pinch, fold, wedge inclusions, collapses and wedge maps; the homotopy axiom applies to based homotopies between these cellular endpoints. Let $r$ be the geometric degree in [F1]. Then $[f]=r[\mathrm{i}\mathrm{d}]$ under pinch addition. We compare $r$ with the homological degree $d$ below.

Write $P:S^p\to S^p\vee S^p$ for the pinch and $F:S^p\vee S^p\to S^p$ for the fold, so that for based self-maps $\alpha ,\beta$ the pinch sum is represented by $F\circ (\alpha \vee \beta )\circ P$; the two components ${\pi }_{1}P,{\pi }_{2}P$ are based degree-one maps, hence are based homotopic to the identity.

For $p=0$ the only based self-maps are identity and collapse. Identity induces identity by functoriality; collapse factors through a point, whose reduced groups are zero by the empty-wedge clauses of [F3] and [F4], so it induces zero. These are also the integers acting on reduced $H_0(S^0;\mathbb{Z})$ by [F6]. No degree is defined here using unreduced $H_0(S^0;\mathbb{Z})\cong {\mathbb{Z}}^2$.

For homology, $P_{\ast }:{\tilde{h}}_n(S^p)\to {\tilde{h}}_n(S^p\vee S^p)$ has components $({\pi }_{1}P{)}_{\ast }=({\pi }_{2}P{)}_{\ast }=\mathrm{i}\mathrm{d}$ under the wedge isomorphism of [F3], hence is the diagonal; and $F_{\ast }$ is the sum map because $F$ restricts to the identity on each summand. Therefore $(F\circ (\alpha \vee \beta )\circ P{)}_{\ast }={\alpha }_{\ast }+{\beta }_{\ast }$ for all based $\alpha ,\beta$.

For cohomology, $F^{\ast }$ is the diagonal and $P^{\ast }$ is the sum map under the wedge isomorphism of [F4], because $F$ restricts to the identity on each summand and the components of $P$ are degree one; therefore $(F\circ (\alpha \vee \beta )\circ P{)}^{\ast }={\alpha }^{\ast }+{\beta }^{\ast }$ for all based $\alpha ,\beta$.

By step 1.1 and functoriality of $\tilde{h}$, $f_{\ast }$ equals the $r$-fold sum of ${\mathrm{i}\mathrm{d}}_{\ast }$ in the group $\mathrm{End}({\tilde{h}}_n(S^p))$, which is multiplication by $r$ by step 2.1. For negative $r$, an inverse class has the negative induced map, since adding it to the original class gives a nullhomotopic map, which factors through the zero reduced group of a point. The same observation includes $r=0$.

By step 1.1 and functoriality of $\tilde{g}$, $f^{\ast }$ equals the $r$-fold sum of ${\mathrm{i}\mathrm{d}}^{\ast }$ in the group $\mathrm{End}({\tilde{g}}^n(S^p))$, which is multiplication by $r$ by step 2.2, including $r<0$: the additive inverse class induces the negative endomorphism since its pinch sum with the original is null and hence induces zero through a point.

The same finite pinch calculation in ordinary integral homology identifies the two degree conventions. By [F6], the top homology of a finite wedge of $p$-spheres is free on the sphere inclusions: its cellular complex has a common vertex, one generator per $p$-cell, and zero boundary (also for $p=1$, since both endpoints attach to that vertex). The collapses give the corresponding coordinate projections. Thus pinch is diagonal and fold is addition in top homology, exactly as in step 2.1. Homotopy invariance [F6] and $[f]=r[\mathrm{i}\mathrm{d}]$ give $f_{\ast }[S^p]=r[S^p]$. By the definition [F2], $d=r$.

Combining steps 1.3, 3.1 and 3.2 gives the asserted multiplication by the degree on every reduced homology and cohomology group of $S^p$.

If $X=\mathrm{\varnothing }$, all skeletal pairs are $(\mathrm{\varnothing },\mathrm{\varnothing })$. Their based cofiber is the cofiber of $\ast \to \ast$, hence a point, and [F3] gives $E_1^{p,q}=0$ for every $p,q$. There are no cells, so [F4] gives the zero Hom group. For any $X$ and $p<0$ the same argument applies. These cases are settled without taking the unbased quotient $\mathrm{\varnothing }\mathrm{/}\mathrm{\varnothing }$.

Suppose now $X\mathrm{\ne }\mathrm{\varnothing }$ and $p\ge 0$. For $p=0$, the cofiber of $\ast \hookrightarrow X_{+}^0$ is $X_{+}^0$ itself, which is the finite wedge of one $S^0$ for each vertex. For $p\ge 1$, $X^{p-1}$ is nonempty since $X$ has a vertex. By [F1] the cofiber of $X_{+}^{p-1}\hookrightarrow X_{+}^p$ is based homotopy equivalent to $X_{+}^p\mathrm{/}{X}_{+}^{p-1}=X^p\mathrm{/}{X}^{p-1}$. This is a wedge of $p$-spheres: each supplied characteristic disk has its whole boundary collapsed to the single quotient vertex, and the quotient CW topology of [F2] is exactly the wedge topology. If there are no $p$-cells the quotient is a point. All comparisons commute with the quotient maps induced by cellular maps.

We verify the degree action needed for cellular naturality directly. For $p\ge 1$ let $P:S^p\to S^p\vee S^p$ be the oriented pinch and $r_1,r_2$ its two collapse projections. Each $r_{i}P$ is homotopic to identity (shrink the collapsed half of the sphere, or use its degree $1$ and [F5]). Under the wedge isomorphism [F3], a class with coordinates $(a,b)$ is $r_1^{\ast }a+r_2^{\ast }b$, since its two restrictions are $a,b$ and the off-diagonal composites are constant. Thus $P^{\ast }(a,b)=a+b$. For based self-maps $f,g$, their pinch sum therefore induces $(f+g{)}^{\ast }=f^{\ast }+g^{\ast }$. Constant maps induce zero because they factor through the point. By [F5], a degree-$d$ class is the $d$-fold sum of the identity class; the inverse class has pullback $-\mathrm{i}\mathrm{d}$ because its sum with identity is null. Hence every degree-$d$ self-map acts as multiplication by $d$, for positive, zero and negative $d$, in every reduced degree.

Apply [F3] to step 1.2. Restriction to the finite wedge summands followed by inverse iterated suspension gives $E_1^{p,q}\cong {\prod }_{\alpha \in I_p}{h}^q(\ast )$. A homomorphism from the free cellular chain group is uniquely its values on the oriented cell basis [F4], so this is the required Hom group. Choose each sphere coordinate to match its oriented cell; independence of an orientation-preserving sphere coordinate change follows from [F5], since a degree-one self-map is based homotopic to identity for $p\ge 1$. For vertices use the canonical generator of each singleton; if a negative generator was supplied, negate its coefficient coordinate. Empty cell sets give zero groups on both sides.

Let $f:X\to Y$ be cellular. For $p\ge 1$ its skeletal quotient map is a based map between the sphere wedges of step 1.2. Under [F3], the matrix component from target cell $\beta$ to source cell $\alpha$ is the pullback of $r_{\beta }\bar{f}{\iota }_{\alpha }:S^p\to S^p$, where $\iota$ includes a summand and $r$ collapses the others: the insertion of one coefficient is $r_{\beta }^{\ast }$ and extraction is ${\iota }_{\alpha }^{\ast }$. By step 1.3 this component is multiplication by the integer degree $d_{\beta \alpha }$. The same integers are the cellular chain matrix of $f$, since those chains are relative homology [F4], and on the sphere quotients the matrix components are the same inclusions and collapses. To compare degree conventions explicitly, in ordinary reduced homology the pinch sends the sphere generator to the pair of generators and the fold adds them; thus a pinch sum acts by the sum of the integers, and identity acts by $1$. The classification [F5] therefore makes its integer degree exactly its action on top homology. The Hom pullback is therefore $(a_{\beta }{)}_{\beta }\mapsto ({\sum }_{\beta }{d}_{\beta \alpha }{a}_{\beta }{)}_{\alpha }$, exactly the reduced-theory pullback. For $p=0$, each vertex maps to a vertex and both formulas simply copy its target coordinate; changing chosen vertex signs conjugates both matrices by the same signs. Empty source, empty target when a map exists, and empty cell sets give the zero maps where appropriate.

A morphism of reduced theories commutes with maps and suspension by definition [F3]; hence it commutes with summand restriction and the inverse suspension identifications of step 2.1, acting on each coordinate by its homomorphism $h^q(\ast )\to k^q(\ast )$. Together with step 3.1 this proves both stated naturalities. Above the dimension of $X$ the skeletal quotient is a point and there are no cells. No dimension axiom, restriction on $q$, or compatibility between separately supplied connecting maps and suspension is needed for this first-page group calculation.

The differential is $d_1=j\circ k=jk$ in the exact-couple indexing: a class in $E_1^{p,q}=h^{p+q}(X^p,X^{p-1})$ is first sent by the pair map $k$ to $h^{p+q}(X^p)$, and then by the connecting map $j$ of the next pair to $h^{p+q+1}(X^{p+1},X^p)=E_1^{p+1,q}$.

Suppose $p\ge 1$. Restricting along the characteristic map ${\phi }_{\tau }:(D^{p+1},\mathrm{\partial }{D}^{p+1})\to (X^{p+1},X^p)$ and collapsing the complement of a $p$-cell $\sigma$ exhibits the $(\tau ,\sigma )$ component of $d_1$ as the map induced by $c_{\sigma }\circ \pi \circ {\phi }_{\tau }{\mathrm{\mid }}_{\mathrm{\partial }{D}^{p+1}}:S^p\to S^p$. The disk-boundary connector agrees with positive suspension in these oriented coordinates: cap the oriented disk with the cone on its outward-oriented boundary. The cone has orientation $dt$ followed by the boundary orientation, so its base boundary is the negative of the disk boundary. They glue to an oriented sphere, and collapse of the disk maps the cone to the positively oriented suspension. Thus no extra sign enters the component diagram. To type the characteristic pair map, first approximate its boundary cellularly in $X^p$, extend that boundary homotopy over the disk by [F7], and approximate the disk rel its now cellular boundary. All sources are finite, and the resulting map is homotopic through pair maps to the original; cellular representatives determine the same maps on the cofiber groups by homotopy invariance. For $p\ge 1$ the boundary vertex now maps into $X^0\subseteq X^{p-1}$, so the attaching-sphere composite after collapse is based. Its degree is unchanged. This is the attaching map followed by collapse onto the $\sigma$-sphere, as in the source diagram.

Suppose $p=0$ and orient a one-cell $e_{\tau }^1$ from its initial endpoint $v_{-}$ to its terminal endpoint $v_{+}$. Naturality reduces its component of $d_1$ to the boundary for $(D^1,S^0)$. The cofiber of $(S^0{)}_{+}\hookrightarrow (D^1{)}_{+}$ is a graph consisting of the oriented edge and a spoke from each endpoint to the common cone tip. Its oriented cycle traverses the edge from initial to terminal, then the terminal spoke forwards, then the initial spoke backwards. Collapsing the edge gives the map to $\mathrm{\Sigma }(S^0{)}_{+}$ with degrees $+1$ on the terminal circle and $-1$ on the initial circle. Wedge coordinates, [F3] in dimension one, and the normalized connector [F6] therefore send the endpoint coefficients to $a(v_{+})-a(v_{-})$; hence its coefficient at a vertex $v$ is ${\mathbf{1}}_{\{v_{+}=v\}}-{\mathbf{1}}_{\{v_{-}=v\}}$, including zero for a loop.

The cellular coboundary with coefficients $h^q(\ast )$ has, in the dual cell bases, the incidence entries $[e_{\tau }^{p+1}:e_{\sigma }^p]$ by duality of the cellular boundary.

For $p\ge 1$, the degree action [F3] says that the $(\tau ,\sigma )$ component of $d_1$ is multiplication by the degree of the map in step 1.2, namely the incidence number $[e_{\tau }^{p+1}:e_{\sigma }^p]$.