nLab Cartan-Eilenberg spectral sequence

Idea

In (Cartan-Eilenberg 56, XV.7) axioms for certain systems of bigraded modules $H(p,q)$ (“Cartan-Eilenberg systems”) are given which imply the existence of a spectral sequence converging to $H(-\infty,\infty)$. See also (Switzer 75, section 15, 1-7).

Definition

A Cartan-Eilenberg system $(H,\eta,\partial)$ consists of modules $H(p,q)$ for each $p\le q$, morphisms $\eta\colon H(p',q')\to H(p,q)$ for all $p\le p'$, $q\le q'$, and boundary morphisms $\partial\colon H(p,q) \to H(q,r)$ for all $p\le q\le r$, such that

1. $\eta=\mathrm{id}\colon H(p,q)\to H(p,q)$,

2. $\eta=\eta\circ\eta\colon H(p'',q'')\to H(p',q')\to H(p,q)$,

3. $\eta$ and $\partial$ commute,

4. there are long exact sequences $\cdots\to H(q,r)\stackrel\eta\to H(p,r)\stackrel\eta\to H(p,q)\stackrel\partial\to H(q,r)\to\cdots$.

Subject to convergence conditions…

The spectral sequence induced from $(H,\eta,\partial)$, is defined by

$$Z^r_p=\mathrm{im}\bigl(H(p,p+r)\stackrel\eta\to H(p,p+1)\bigr)\;,$$

$$B^r_p=\mathrm{im}\bigl(H(p-r+1,p)\stackrel\partial\to H(p,p+1)\bigr)\;,$$

$$E^r_p=Z^r_p/B^r_p\;,$$

$$d^r_p\colon Z^r_p/B^r_p\twoheadrightarrow Z^r_p/Z^{r+1}_p\cong B^{r+1}_{p+r}/B^r_{p+r}\hookrightarrow Z^r_{p+r}/B^r_{p+r}\;.$$

In particular

$$\ker(d^r_p)=Z^{r+1}_p/B^r_p\qquad\text{and}\qquad \mathrm{im}(d^r_p)=B^{r+1}_p/B^r_p\;.$$

For $a=\eta(a_0)\in H(p,p+1)$, $a_0\in H(p,p+r)$, one has

$$d^r_p([a])=[\partial a_0]\in E^r_p\qquad\text{with}\qquad\partial a_0\in H(p+r,p+r+1)\;.$$

Examples

Atiyah-Hirzebruch spectral sequences

The key example of such a system are the relative cohomology groups $E^\bullet(X^q, X^p)$ of a filtered topological space (see at spectral sequence of a filtered complex) for $E$ any generalized cohomology theory (Cartan-Eilenberg 56, XV.7, Example 2). That $E^\bullet(X^q, X^p)$ forms a suitable system of modules is the statement of the exact sequence for triples of any generalized cohomology theory.

This case of the Cartan-Eilenberg spectral sequence came to be known as the Atiyah-Hirzebruch spectral sequence.

Definition

For $\pi\colon X\to B$ a Serre fibration over a CW-complex $B$. And for $(\tilde h^\bullet,\delta,\wedge)$ a multiplicative reduced generalized (Eilenberg-Steenrod) cohomology theory.

define a Cartan-Eilenberg system $(H,\eta,\partial)$ by

$$H(p,q)=\tilde h^\bullet(X^{q-1}/X^{p-1})$$

(where $X^k=\pi^{-1}(B^k)$) for $p\le q$ with the obvious maps $\eta\colon H(p',q')\to H(p,q)$ for $p\le p'$, $q\le q'$.

The Cartan-Eilenberg spectral sequence of this Cartan-Eilenberg system is the Serre-Atiyah-Hirzebruch spectral sequence.

The corresponding long exact sequences take the form

$$\cdots\to\tilde h^\bullet(X^{r-1},X^{q-1})\to\tilde h^\bullet(X^{r-1},X^{p-1}) \to\tilde h^\bullet(X^{q-1},X^{p-1})\stackrel\delta\to \tilde h^\bullet(X^{r-1},X^{q-1})\to\cdots$$

Related concepts

• multiplicative spectral sequence

References

• Henri Cartan, Samuel Eilenberg, Homological algebra, Princeton Univ. Press (1956)

• Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.