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Cohomological Atiyah–Hirzebruch spectral sequence

Statement

Let $X$ be a finite CW complex, let $\tilde{h}$ be a reduced generalized cohomology theory, and let $h$ be the associated CW-pair theory (Reduced and unreduced generalized cohomology theories correspond). There is a cohomological spectral sequence with $E_{1}^{p, q} = h^{p + q} (X^{p}, X^{p - 1}), E_{2}^{p, q} ≅ H^{p} (X; h^{q} (*)), d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1},$ whose stable page is the associated graded of the skeletal filtration of Skeletal filtration for generalized cohomology: $E_{\infty}^{p, q} ≅ F^{p} h^{p + q} (X) / F^{p + 1} h^{p + q} (X) .$ The spectral sequence is natural in $X$ for cellular maps and in the theory $\tilde{h}$ for morphisms of reduced generalized cohomology theories.

Facts & Assumptions

[F1]

The pair theory realizes $h^{n} (X, A)$ with a natural long exact sequence and connecting maps, with $h^{n} (X) = h^{n} (X, \emptyset)$ and $h^{n} (X, A) ≅ {\tilde{h}}^{n} (X / A)$ for nonempty $A$ (Reduced and unreduced generalized cohomology theories correspond).

[F2]

A page- $r$ homological exact couple consists of bigraded families with maps $i : D_{p, q} \to D_{p + 1, q - 1}$ , $j : D_{p, q} \to E_{p + 1 - r, q + r - 1}$ and $k : E_{p, q} \to D_{p - 1, q}$ exact at all three vertices; an initial exact couple gives a spectral sequence starting at $E^{1} = E$ with differentials of bidegree $(- r, r - 1)$ , subquotient description and local-lift formula by An exact couple generates a spectral sequence (Exact couple).

[F3]

The first page is cellular cochains and the first differential is the cellular coboundary, so the second page is $H^{p} (X; h^{q} (*))$ (The AHSS E-one page is cellular cochains with theory coefficients, The AHSS first differential is the cellular coboundary).

[F4]

A cohomological spectral sequence is a homological spectral sequence under $E_{r}^{p, q} : = E_{- p, - q}^{r}$ , with differentials of bidegree $(r, 1 - r)$ (Cohomological spectral sequence).

[F5]

The skeletal filtration satisfies $F^{p + 1} \subseteq F^{p}$ , $F^{p} = \ker (h^{n} (X) \to h^{n} (X^{p - 1}))$ , $F^{p + 1} = \ker (h^{n} (X) \to h^{n} (X^{p}))$ , $F^{p} = im (h^{n} (X, X^{p - 1}) \to h^{n} (X))$ , and it is exhaustive and bounded for finite $X$ (Skeletal filtration for generalized cohomology).

Proof

technique · direct

Given: A finite CW complex $X$ of dimension $N$ , a reduced generalized cohomology theory $\tilde{h}$ , the associated pair theory $h$ , and the conventions $X^{p} = \emptyset$ for $p < 0$ , $X^{p} = X$ for $p \geq N$ .

1.1

Put $D_{p, q} : = h^{- p - q - 1} (X^{- p - 1})$ and $E_{p, q} : = h^{- p - q} (X^{- p}, X^{- p - 1})$ , and let $i$ be restriction from $X^{- p - 1}$ to $X^{- p - 2}$ , $j$ the connecting map of the pair $(X^{- p}, X^{- p - 1})$ , and $k$ the pair map to $h^{- p - q} (X^{- p})$ . The three exactness conditions are the corresponding portions of the pair long exact sequences: at $D_{p, q}$ the incoming restriction is $h^{- p - q - 1} (X^{- p}) \to h^{- p - q - 1} (X^{- p - 1})$ and its image is the kernel of $j$ ; the image of $j$ is the kernel of the pair map; and the image of the pair map is the kernel of the outgoing restriction to $X^{- p - 1}$ . Hence the data form an initial exact couple in the sense of [F2].

F1F2given

1.2

The identification $(P, Q) = (- p, - q)$ turns the pair $(X^{- p}, X^{- p - 1})$ into $(X^{P}, X^{P - 1})$ and the group $E_{p, q} = h^{- p - q} (X^{- p}, X^{- p - 1})$ into $h^{P + Q} (X^{P}, X^{P - 1})$ ; under this reindexing the exact-couple differentials $d^{r} : E_{p, q} \to E_{p - r, q + r - 1}$ become $d_{r} : E_{r}^{P, Q} \to E_{r}^{P + r, Q - r + 1}$ .

F4given

2.1

Applying [F2] to the couple of step 1.1 produces a homological spectral sequence with $E^{1} = E$ , differentials of bidegree $(- r, r - 1)$ , subquotients $E_{p, q}^{r} ≅ N_{p, q}^{r} / B_{p, q}^{r}$ and the local-lift formula $k e = i^{r - 1} x \Rightarrow d^{r} [e] = [j x]$ .

F2step 1.1

2.2

The page- $1$ terms are the cellular cochains and the page- $1$ differential is the cellular coboundary, so applying the reindexing of step 1.2 to the second page gives $E_{2}^{P, Q} ≅ H^{P} (X; h^{Q} (*))$ .

F3step 1.1step 1.2

3.1

By definition of a cohomological spectral sequence, the reindexed data $E_{r}^{P, Q} : = E_{- P, - Q}^{r}$ form a cohomological spectral sequence whose differentials have bidegree $(P, Q) \mapsto (P + r, Q - r + 1)$ and whose second page is $H^{P} (X; h^{Q} (*))$ , as required.

F4step 2.1step 2.2

3.2

For the convergence, fix $(p, q)$ and put $P = - p$ , $Q = - q$ , $n = P + Q = - p - q$ . The numerators $N_{p, q}^{r} = k^{- 1} (im (i^{r - 1} : D_{p - r, q + r - 1} \to D_{p - 1, q}))$ equal $k^{- 1} (im (h^{n} (X^{P + r - 1}) \to h^{n} (X^{P})))$ , so they decrease with $r$ and stabilize at $N_{p, q}^{\infty} = k^{- 1} (im (h^{n} (X) \to h^{n} (X^{P})))$ once $X^{P + r - 1} = X$ . The denominators $B_{p, q}^{r} = j (\ker (i^{r - 1} : D_{p, q} \to D_{p + r - 1, q - r + 1}))$ equal $j (\ker (h^{n - 1} (X^{P - 1}) \to h^{n - 1} (X^{P - r})))$ , so they increase and, for $r > P$ , equal $j (h^{n - 1} (X^{P - 1})) = \ker (k)$ because $X^{P - r} = \emptyset$ has zero cohomology.

F1F2F5step 2.1

4.1

The map $k$ sends the stable numerator onto $K : = im (h^{n} (X) \to h^{n} (X^{P})) \cap \ker (h^{n} (X^{P}) \to h^{n} (X^{P - 1}))$ with kernel $B_{p, q}^{\infty} = \ker (k)$ , so $E_{p, q}^{\infty} = N_{p, q}^{\infty} / B_{p, q}^{\infty} ≅ K$ . Restriction $h^{n} (X) \to h^{n} (X^{P})$ carries $\ker (h^{n} (X) \to h^{n} (X^{P - 1}))$ onto $K$ with kernel $\ker (h^{n} (X) \to h^{n} (X^{P}))$ ; hence $E_{p, q}^{\infty} ≅ F^{P} h^{n} (X) / F^{P + 1} h^{n} (X)$ .

F1F5step 3.2

5.1

The reindexed spectral sequence of step 3.1 has the asserted first and second pages and differentials, and step 4.1 identifies its stable terms with the associated graded of the skeletal filtration. A cellular map preserves every skeleton and therefore gives commuting maps of all the pair sequences used in step 1.1; a morphism of reduced theories gives the same commuting ladders objectwise. These ladders are morphisms of exact couples, so A map of exact couples induces a map of spectral sequences supplies the asserted natural page maps.

step 1.1step 3.1step 4.1∎

Source notes

Compare Loizides, §3, Theorems 3.2 and 3.4 with Lemma 3.6, printed pp. 4–8, for the exact couple, the E-two page $H^{p} (X; h^{q})$ and the identification of $E_{\infty}$ with the associated graded of $\ker (h^{n} (X) \to h^{n} (X^{m}))$ ; and Ji, §§3.1–3.2, printed pp. 10–12, for the right-half-plane support and collapse computations. The indexing here is chosen so that $E_{2}^{p, q} = H^{p} (X; h^{q} (*))$ and $d_{r}$ has bidegree $(r, 1 - r)$ .

Depends on

Used by

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Sources

Yiannis Loizides, The Atiyah–Hirzebruch Spectral Sequence, §3, Theorems 3.2 and 3.4, printed pp. 4–8 (standard reference, not scraped)
Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, §§3.1–3.2, printed pp. 10–12 (standard reference, not scraped)