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Homological AHSS exact couple from the skeletal filtration

Statement

Let $X$ be a finite CW complex with chosen cells and orientations and let $\tilde{h}$ be a reduced generalized homology theory. Set $D_{p, q} : = h_{p + q} (X^{p}), E_{p, q} : = h_{p + q} (X^{p}, X^{p - 1}),$ with $h_{n} (X, A)$ the pair groups associated with $\tilde{h}$ and the convention $X^{p} = \emptyset$ for $p < 0$ . Let $i_{p, q} : D_{p, q} \to D_{p + 1, q - 1}, j_{p, q} : D_{p, q} \to E_{p, q}, k_{p, q} : E_{p, q} \to D_{p - 1, q}$ be induced respectively by the inclusion $X^{p} ↪ X^{p + 1}$ , by the pair map with respect to $(X^{p}, X^{p - 1})$ , and by the connecting map of that pair. Then $(D, E, i, j, k)$ is an initial exact couple in the sense of Exact couple, its first page is $E_{p, q}^{1} = h_{p + q} (X^{p}, X^{p - 1}) ≅ C_{p}^{c e l l} (X; h_{q} (*)),$ and its first differential $d_{1} = j \circ k$ is the cellular boundary with the incidence-degree matrix. In particular $E_{p, q}^{2} ≅ H_{p} (X; h_{q} (*))$ . Here the displayed identification of each sphere summand with $h_{q} (*)$ is suspension-normalized: it uses the oriented quotient sphere and the structural suspension isomorphisms of the theory. The pair boundary is the cofiber map followed by inverse structural suspension, so the same normalization is used in the calculation of $d_{1}$ .

Facts & Assumptions

[F1]

A reduced generalized homology theory has based homotopy invariance, suspension-compatible cofiber exact sequences, and for a finite wedge an isomorphism $⨁_{α} {\tilde{h}}_{n} (S_{α}^{p}) ≅ {\tilde{h}}_{n} (⋁_{α} S_{α}^{p})$ (Reduced generalized homology theory).

[F2]

For a reduced generalized homology theory define the pair groups of a CW pair $(X, A)$ by $h_{n} (X, A) : = {\tilde{h}}_{n} (C_{i})$ for the based inclusion $i : A_{+} \to X_{+}$ , and $h_{n} (X, \emptyset) : = {\tilde{h}}_{n} (X_{+})$ . For $A \neq \emptyset$ a CW subcomplex inclusion is a based cofibration and the collapse $C_{i} \to X / A$ is a based homotopy equivalence, so $h_{n} (X, A) ≅ {\tilde{h}}_{n} (X / A)$ . The cofiber exact sequence is the pair long exact sequence, and its natural boundary is the structural cofiber map followed by inverse suspension (Reduced generalized homology theory, Unreduced pair and reduced quotient axioms are equivalent on cw pairs, Relative CW inclusions are cofibrations, Cofiber of a based cofibration is equivalent to the quotient).

[F3]

For $p \geq 1$ the quotient $X^{p} / X^{p - 1}$ is the finite wedge of the $p$ -spheres belonging to the $p$ -cells, and $X^{0}$ is a finite discrete set (CW quotients and collapse of a contractible subcomplex, The wedge of a family of pointed spaces).

[F4]

The chosen orientation $D^{p} / S^{p - 1} ≅ S^{p}$ , followed by the $p$ -fold structural suspension isomorphism, gives the canonical coefficient identification ${\tilde{h}}_{p + q} (S^{p}) ≅ h_{q} (*)$ of Coefficient groups of a generalized homology theory. For the disk pair, the pair boundary has target $h_{n - 1} (S^{p - 1}) = {\tilde{h}}_{n - 1} ((S^{p - 1})_{+})$ , not merely ${\tilde{h}}_{n - 1} (S^{p - 1})$ . Its image is the kernel of $h_{n - 1} (S^{p - 1}) \to h_{n - 1} (D^{p})$ ; under the usual based-sphere splitting this kernel is the reduced sphere summand, and [F2] identifies the induced map onto that summand with inverse structural suspension. No isomorphism onto the whole unreduced target is asserted.

[F5]

A based degree- $d$ map $S^{p} \to S^{p}$ induces multiplication by $d$ on every reduced generalized homology group (Degree-d sphere maps act by multiplication by d in any generalized theory).

[F6]

The cellular $p$ -chains are free on the oriented $p$ -cells, the cellular boundary is the incidence-degree matrix, and cellular homology computes singular homology. The incidence number $[e_{σ}^{p} : e_{τ}^{p - 1}]$ is the degree of the composite of the attaching map of $e_{σ}^{p}$ with the collapse onto the $(p - 1)$ -sphere $e_{τ}^{p - 1}$ ; the cellular boundary is the incidence-degree matrix (Oriented cellular chain group, Incidence number of two CW cells, Cellular boundary is the incidence degree matrix, Cellular homology computes singular homology).

Proof

technique · direct

Given: A finite CW complex $X$ with chosen cells and orientations, a reduced generalized homology theory $\tilde{h}$ , and the groups and maps displayed in the statement.

1.1

The exactness conditions of Exact couple hold with $r = 1$ : the image of $i_{p - 1, q + 1}$ equals the kernel of $j_{p, q}$ by exactness of the pair sequence of $(X^{p}, X^{p - 1})$ at $h_{p + q} (X^{p})$ ; the image of $j_{p, q}$ equals the kernel of $k_{p, q}$ by exactness of that same sequence at $E_{p, q} = h_{p + q} (X^{p}, X^{p - 1})$ ; and the image of $k_{p + 1, q}$ equals the kernel of $i_{p, q}$ by exactness of the pair sequence of $(X^{p + 1}, X^{p})$ at $h_{p + q} (X^{p})$ .

F2given

1.2

The wedge decomposition [F3] and the quotient identification of [F2] give $E_{p, q} ≅ {\tilde{h}}_{p + q} (⋁_{α \in I_{p}} S_{α}^{p})$ . The wedge axiom [F1], followed on every oriented sphere summand by the boundary-normalized identification of [F4], gives ${\tilde{h}}_{p + q} (⋁_{α \in I_{p}} S_{α}^{p}) ≅ ⨁_{α \in I_{p}} h_{q} (*)$ .

F1F2F3F4

1.3

The right-hand side $⨁_{α \in I_{p}} h_{q} (*)$ is the free cellular chain group $C_{p}^{c e l l} (X; h_{q} (*)) = ⨁_{α \in I_{p}} h_{q} (*)$ on the oriented $p$ -cells, and the cellular boundary sends the $σ$ -summand to the sum over $(p - 1)$ -cells of the incidence numbers.

1.4

The differential is $d_{1} = j \circ k$ . By naturality of the pair boundary for the characteristic map $(D^{p}, S^{p - 1}) \to (X^{p}, X^{p - 1})$ and the cofiber-boundary formula in [F2], the $σ$ -summand first maps by the attaching map into $h_{p + q - 1} (X^{p - 1})$ . Projecting with $j$ to the $τ$ -summand composes the attaching map with the collapse $X^{p - 1} \to X^{p - 1} / X^{p - 2}$ and then with the projection to the sphere of $e_{τ}^{p - 1}$ . Under the suspension-normalized coordinates of [F4], the resulting coefficient homomorphism is the map induced by that attaching-and-collapse self-map of $S^{p - 1}$ ; this uses compatibility of the pair boundary with suspension, not an isomorphism from the disk pair group onto all of $h_{p + q - 1} (S^{p - 1})$ .

F2F4given

2.1

For $p \geq 2$ , the suspension-normalized comparison in step 1.4 reduces the $(τ, σ)$ component to the self-map of $S^{p - 1}$ defining the incidence number; [F5] therefore makes it multiplication by $[e_{σ}^{p} : e_{τ}^{p - 1}]$ . For $p = 1$ , the cofiber-boundary convention of [F2] for an oriented characteristic interval gives terminal endpoint minus initial endpoint, so the two possible $0$ -cell projections have coefficients $+ 1$ and $- 1$ . For $p = 0$ the target is zero. Thus [F6] identifies $d_{1}$ with the cellular boundary in every dimension.

F2F4F5F6step 1.4

3.1

Steps 1.1, 1.2 and 2.1 exhibit an initial exact couple whose first page is the cellular chain complex with coefficients $h_{q} (*)$ and whose first differential is the cellular boundary; hence $E_{p, q}^{2} ≅ H_{p} (X; h_{q} (*))$ by definition of the second page and the identification of cellular with singular homology.

F6step 1.1step 1.2step 2.1

4.1

This proves the asserted exact couple, first page, differential and second page.

step 3.1∎

Source notes

Compare Davis–Kirk, §8.8 and Theorem 9.6, printed pp. 227–246, for the skeletal exact couple of a generalized homology theory, the identification of the first page with cellular chains and the bidegree of the differential.

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Sources

Davis–Kirk, Lecture Notes in Algebraic Topology, §8.8 and Theorem 9.6, printed pp. 227–246 (standard reference, not scraped)