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Cartan-Eilenberg spectral sequence

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Homological algebra

homological algebra

(also nonabelian homological algebra)

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diagram chasing

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Idea

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

Definition

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

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

  2. η=ηη:H(p,q)H(p,q)H(p,q)\eta=\eta\circ\eta\colon H(p'',q'')\to H(p',q')\to H(p,q),

  3. η\eta and \partial commute,

  4. there are long exact sequences H(q,r)ηH(p,r)ηH(p,q)H(q,r)\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,η,)(H,\eta,\partial), is defined by

Z p r=im(H(p,p+r)ηH(p,p+1)), Z^r_p=\mathrm{im}\bigl(H(p,p+r)\stackrel\eta\to H(p,p+1)\bigr)\;,
B p r=im(H(pr+1,p)H(p,p+1)), B^r_p=\mathrm{im}\bigl(H(p-r+1,p)\stackrel\partial\to H(p,p+1)\bigr)\;,
E p r=Z p r/B p r, E^r_p=Z^r_p/B^r_p\;,
d p r:Z p r/B p rZ p r/Z p r+1B p+r r+1/B p+r rZ p+r r/B p+r r. 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 p r)=Z p r+1/B p randim(d p r)=B p r+1/B p r.\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=η(a 0)H(p,p+1)a=\eta(a_0)\in H(p,p+1), a 0H(p,p+r)a_0\in H(p,p+r), one has

d p r([a])=[a 0]E p rwitha 0H(p+r,p+r+1).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 (X q,X p)E^\bullet(X^q, X^p) of a filtered topological space (see at spectral sequence of a filtered complex) for EE any generalized cohomology theory (Cartan-Eilenberg 56, XV.7, Example 2). That E (X q,X p)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 π:XB\pi\colon X\to B a Serre fibration over a CW-complex BB. And for (h˜ ,δ,)(\tilde h^\bullet,\delta,\wedge) a multiplicative reduced generalized (Eilenberg-Steenrod) cohomology theory.

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

H(p,q)=h˜ (X q1/X p1) H(p,q)=\tilde h^\bullet(X^{q-1}/X^{p-1})

(where X k=π 1(B k)X^k=\pi^{-1}(B^k)) for pqp\le q with the obvious maps η:H(p,q)H(p,q)\eta\colon H(p',q')\to H(p,q) for ppp\le p', qqq\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

h˜ (X r1,X q1)h˜ (X r1,X p1)h˜ (X q1,X p1)δh˜ (X r1,X q1)\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

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.

Last revised on February 29, 2016 at 11:58:55. See the history of this page for a list of all contributions to it.