nLab multiplicative spectral sequence

1. Idea

A spectral sequence is called multiplicative or a spectral ring if there is a bi-graded algebra structure on each page such that the differentials act as graded derivations of total degree 1.

For example the Serre-Atiyah-Hirzebruch spectral sequence with coefficients in a ring spectrum.

2. Constructions

From spectral products on Cartan-Eilenberg systems

The following gives sufficient conditions for a Cartan-Eilenberg spectral sequence to be multiplicative. This is due to (Douady 58). The following is taken from (Goette 15a).

Definition 2.1. Let $(H,\eta,\partial)$, $(H',\eta',\partial')$ und $(H'',\eta'',\partial'')$ be Cartan-Eilenberg systems. A spectral product $\mu\colon(H',\partial')\times(H'',\partial'')\to(H,\partial)$ is a sequence of homomorphisms

$$\mu_r\colon H'(m,m+r)\otimes H''(n,n+r)\to H(m+n,m+n+r)$$

such that for all $m$, $n$, $r\ge 1$, the following two diagrams commute:

$$\array{ H'(m, m+r) \otimes H''(n, n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n, m+n+r) \\ \downarrow^{\mathrlap{\eta' \oplus \eta''}} && \downarrow^{\mathrlap{\eta}} \\ H'(m, m+1) \otimes H''(n, n+1) &\stackrel{\mu_1}{\longrightarrow}& H(m+n, m+n+1) }$$

and

$$\array{ H'(m, m+r) \otimes H''(n, n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n,m+n+r) \\ \downarrow^{\mathrlap{\partial' \otimes \eta'' \oplus \eta' \otimes \partial''}} && \downarrow^{\mathrlap{\partial}} \\ H'(m+r, m+r+1) \otimes H''(n,n+1) \\ \oplus &\stackrel{\mu_1 + \mu_1}{\longrightarrow}& H_{p+q-1}(m+n+r, m+n+r+1) \\ H'(m,m+1) \otimes H''(n+r, n+r+1) } \,.$$

Remark 2.2. The first diagram in def. 2.1 is weaker than in (Douady 58). The second may be read as a Leibniz rule.

Write $E$ for the Cartan-Eilenberg spectral sequence induced from the Cartan-Eilenberg system $H$.

Proposition 2.3. A spectral product $\mu\colon(H',\partial')\times(H'',\partial'')\to(H,\partial)$ as in def. 2.1 induces products

$$\mu^r\colon E^{\prime r}_m\otimes E^{\prime\prime r}_n\to E^r_{m+n}\;,$$

such that

1. $\mu^1=\mu_1$

2. $d^r_{m+n}\circ\mu^r=\mu^r\circ(d^{\prime r}_m\otimes\mathrm{id})\pm\mu^r\circ(\mathrm{id}\circ d^{\prime\prime r}_n)$,

3. $\mu^{r+1}$ is induced by $\mu^r$.

(Goette 15a, following Douady 58, theorem II).

Proof. Assume by induction that $\mu^r$ is induced by $\mu_1$. In particular,

$$Z^{\prime r}_m\otimes Z^{\prime\prime r}_n\stackrel{\mu_1}\to Z^r_{m+n}\;,$$

$$B^{\prime r}_m\otimes Z^{\prime\prime r}_n\stackrel{\mu_1}\to B^r_{m+n}\;,$$

$$Z^{\prime r}_m\otimes B^{\prime\prime r}_n\stackrel{\mu_1}\to B^r_{m+n}\;.$$

This is clear for $r=1$ if we put $\mu^1=\mu_1$ because $E^1_p=Z^1_p=H(p,p+1)$ and $B^1_p=0$.

Let $[a]\in Z^{\prime r}_m$, $[b]\in Z^{\prime\prime r}_n$ be represented by $a=\eta'(a_0)\in H'(m,m+1)$, $b=\eta''(b_0)\in H''(n,n+1)$ with $a_0\in H'(m,m+r)$, $b_0\in H''(n,n+r)$. Using the first diagram and the construction of $d^r_{m+n}$, we conclude that

$$(d^r_{m+n}\circ\mu^r)([a]\otimes[b])=d^r_{m+n}[\mu_1(a\otimes b)]=d^r_{m+n}[\eta(\mu_r(a_0\otimes b_0))]=(\partial\circ\mu_r)(a_0\otimes b_0) \;.$$

From the second diagram, we get

$$(\partial\circ\mu_r)(a_0\otimes b_0)=\mu_1(\partial'a_0\otimes\eta''b_0)\pm\mu_1(\eta'a_0\otimes\partial''b_0)=\mu^r(d^{\prime r}_m[a]\otimes[b])\pm\mu^r([a]\otimes d^{\prime\prime r}_n[b]) \;.$$

This proves the Leibniz rule (2).

From the Leibniz rule and the facts that $\ker(d^r_p)=Z^{r+1}_p/B^r_p$ and $\mathrm{im}(d^r_p)=B^{r+1}_p/B^r_p$, we conclude that $\mu^r$ induces a product on $E^{r+1}_p\cong\ker(d^r_p)/\mathrm{im}(d^r_p)$, which proves (3). Because $\mu^r$ is induced by $\mu_1$, so is $\mu^{r+1}$, and we can continue the induction.  ▮

3. Examples

AHSS for multiplicative cohomology

We discuss that the multiplicative structure on the cohomology Serre-Atiyah-Hirzebruch spectral sequence for multiplicative generalized cohomology. This is taken from (Goette 15b).

Definition 3.1. 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.

Definition 3.2. The spectral product $\mu\colon(H,\eta,\partial)\times(H,\eta,\partial)\to(H,\eta,\partial)$, def. 2.1, on the Cartan-Eilenberg system of def. 3.1 is that given by the following morphism

$$\begin{aligned} F_{m,n,r} & \colon (X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1} \cong \bigcup_{a+b=m+n+r-1}(X^a\wedge X^b) / \bigcup_{c+d=m+n-1}(X^c\wedge X^d) \\ & \twoheadrightarrow \bigcup_{a+b=m+n+r-1} (X^a\wedge X^b)/(\bigcup_{a=0}^m(X^{a-1}\wedge X^{m+n+r-a}) \cup\bigcup_{b=0}^n(X^{m+n+r-b}\wedge X^{b-1}) \\ & \cong \bigcup_{a=m+1}^{m+r}(X^{a-1}\wedge X^{m+n+r-a}) / \bigl(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1}\bigr) \\ & \hookrightarrow X^{m+r-1}\wedge X^{n+r-1}/(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1}) \\ & \cong (X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1}) \;. \end{aligned}$$

Together with the diagonal map $\Delta$, for $r\ge 1$, we define

$$\begin{aligned} \mu_r & \colon H(m,m+r)\otimes H(n,n+r) \\ & \cong\tilde h(X^{m+r-1}/X^{m-1})\otimes\tilde h(X^{n+r-1}/X^{n-1}) \\ &\stackrel\wedge\longrightarrow\tilde h\bigl((X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})\bigr) \\ &\stackrel{F_{m,n,r}^*}\longrightarrow\tilde h\bigl((X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1}\bigr) \\ &\stackrel{\Delta_X^*}\longrightarrow\tilde h(X^{m+n+r-1}/X^{m+n-1})=H(m+n,m+n+r)\;. \end{aligned}$$

Proposition 3.3. With def. 3.2, then for all $m$, $n$, $r\ge 1$, the following [[commuting diagram|diagram commutes]]

$$\array{ H(m,m+1) \otimes H(n,n+1) &\stackrel{\mu_1}{\longrightarrow}& H(m+n, m+n+1) \\ \uparrow^{\mathrlap{\eta \oplus \eta}} && \uparrow^{\mathrlap{\eta}} \\ H(m,m+r) \otimes H(n,n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n,m+n+r) \\ \downarrow^{\mathrlap{\partial \otimes \eta \oplus \eta \otimes \partial}} && \downarrow^{\mathrlap{\partial}} \\ H(m+r, m+r+1) \otimes H(n,n+1) \\ \oplus &\stackrel{\mu_1 \pm \mu_1}{\longrightarrow}& H(m+n+r, m+n+r+1) \\ H(m,m+1) \otimes H(n+r, n+r+1) } \,.$$

Hence by prop. 2.3 the spectral product of def. 3.2 defines a mutliplicative structure on the Serre-WhiteheadAtiyah-Hirzebruch spectral sequence for multiplicative generalizted cohomology.

Proof. The upper square commutes because the maps $F_{m,n,r}$ are [[natural transformations]]. For the lower square, we consider the boundary morphism $\delta$ of the triple

$$\begin{aligned} (& X^{m+r}\wedge X^{n+r-1}\cup X^{m+r-1}\wedge X^{n+r}, \\ & X^{m+r}\wedge X^{n-1}\cup X^{m+r-1}\wedge X^{n+r-1}\cup X^{m-1}\wedge X^{n+r}, \\ & X^{m+r}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r}) \end{aligned} \;.$$

The following diagram commutes:

$$\array{ \tilde h^{-p}(X^{m+r-1}/ X^{m-1}) \otimes \tilde h^{-q}(X^{n+r-1}/X^{n-1}) &\stackrel{\wedge}{\longrightarrow}& \tilde h^{-p-q}((X^{m+r-1}/X^{m-1}) \wedge (X^{n+r-1}/X^{n-1}) \\ \downarrow^{\mathrlap{\delta \wedge id \oplus id \wedge \delta}} && \downarrow^{\mathrlap{\delta}} \\ \tilde h(1-p)(X^{m+r}/X^{m+r-1}) \otimes \tilde h^{-q}(X^{n+r-1}/X^{n-1}) && \tilde h^{1-p-q}((X^{m+r}/X^{m+r-1}) \wedge (X^{n+r-1}/X^{n-1})) \\ \oplus &\stackrel{\wedge \oplus \wedge}{\longrightarrow}& \\ \tilde h^{-p}(X^{m+r-1}/X^{m-1}) \otimes \tilde h^{1-q}(X^{n+r}/X^{n+r-1}) && \tilde h^{1-p-q}( (X^{m+r-1}/ X^{m-1}) \wedge (X^{n+r}/ X^{n+r-1}) ) } \,.$$

By extend this diagram to the right using the maps $F_{m,n,r}$ once concludes that the lower square above also commutes.  ▮

4. Related concepts

• [[multiplicative cohomology theory]]

5. References

• [[Adrien Douady]], La suite spectrale d’Adams : structure multiplicative Séminaire Henri Cartan, 11 no. 2 (1958-1959), Exp. No. 19, 13 p (Numdam)

• Brayton Gray, Products in the Atiyah-Hirzebruch spectral sequence and the calculation of $M SO_\ast$, Trans. Amer. Math. Soc. 260 (1980), 475-483 (web)

• [[Stanley Kochmann]], prop. 4.2.9 of [[Bordism, Stable Homotopy and Adams Spectral Sequences]], AMS 1996

• [[John McCleary]], section 2.3 in A User’s Guide to Spectral Sequences, Cambridge University Press (2000)

• [[Daniel Dugger]], Multiplicative structures on homotopy spectral sequences I (arXiv:math/0305173)

• [[Daniel Dugger]], Multiplicative structures on homotopy spectral sequences II (arXiv:math/0305187)

• [[Sebastian Goette]], MO comment a, MO comment b Feb 15, 2015

[[!redirects multiplicative spectral sequences]]