Are there spectral sequences for calculating homology or cohomology of homotopy (co)limits?
Suppose my nice topological space $X$ is the homotopy colimit $$\operatorname{hocolim}D\cong X$$ of a diagram $D\colon I\to \mathbf{Top}$ and the homotopy limit $$\operatorname{holim}E\cong X$$ of a diagram $E\colon J\to \mathbf{Top}$.
Are there spectral sequences relating the singular homology or singular cohomology of $X$ to the (co)homology of the $D(i)$ or the $E(j)$? What if $D$ and $E$ are (co)filtered diagrams? May I replace singular (co)homology by an other (generalized) (co)homology theory?
Solution 1:
- Yes, there is a spectral sequence for which the groups on the $E^1$-page are the singular homology of $D(i)$ for varying $i \in I$, which converges to the singular homology of $X$. This spectral sequence arises from the skeletal filtration $X_n$ of the homotopy colimit $X$: essentially, $X_n$ is the part of the hocolim that comes from $k$-tuples of arrows in $I$ for $k \leq n$.
Let's walk through the construction a bit further, using the usual grading for a spectral sequence of a filtered complex. Assume all our spaces are unbased. Recall that a filtered space $$ X_0 \rightarrow X_1 \rightarrow \ldots \rightarrow X $$ gives an exact couple $$ D^1_{p,q} = H_{p+q}(X_p) $$ $$ E^1_{p,q} = H_{p+q}(X_p,X_{p-1})$$ and that $E^1_{p,q}$ give the groups on the $E^1$-page. By excision this is the homology of the quotient $$ H_{p+q}(X_p/X_{p-1}) $$ and if you examine the construction of the hocolim you see that this quotient is homeomorphic to $$ \left(\coprod_{i_0 \rightarrow \ldots \rightarrow i_p} D(i_0) \times \Delta^p \right) / \left(\coprod_{i_0 \rightarrow \ldots \rightarrow i_p} D(i_0) \times \partial\Delta^p \right) \cong \bigvee_{i_0 \rightarrow \ldots \rightarrow i_p} \ \Sigma^p (D(i_0)_+) $$ Therefore we can rewrite the $E^1$-page $$ E^1_{p,q} \cong \bigoplus_{i_0 \rightarrow \ldots \rightarrow i_p} H_q(D(i_0)) $$ which is zero whenever $q < 0$. Therefore our $E^1$-page is concentrated in the first quadrant! The differentials are the same grading as in the Serre spectral sequence. You can check that this then satisfies the convergence conditions found in e.g. Mosher & Tangora p.66, so the spectral sequence has an $E^\infty$ page that gives the filtration of $H_{p+q}(X)$ by the images of the groups $H_{p+q}(X_p)$.
Finally, the differentials on the $E^1$-page are the alternating sum of the face maps $$ \bigoplus_{i_0 \rightarrow \ldots \rightarrow i_p} H_q(D(i_0)) \overset{d_j}\rightarrow \bigoplus_{i_0 \rightarrow \ldots \rightarrow i_p} H_q(D(i_0)) $$ where $d_j$ deletes the term $i_j$ in the $p$-tuple of arrows and composes the arrows passing thru $i_j$ while doing nothing to $D(i_0)$. That is unless $j = p$, in which case it deletes $i_p$, or $j = 0$, in which case it deletes $i_0$ and applies the arrow $i_0 \rightarrow i_1$ to $D(i_0)$ to land in $D(i_1)$.
This also works for extraordinary homology theories, except for the convergence. If your extraordinary theory is connective (i.e. $H_{n}(*) = 0$ when $n$ is negative) then you should be fine.
This can also be used on cohomology, and everything is fine except that your $E_\infty$ page gives a filtration of $\mathbf{lim}_p H^*(X_p)$, which is not necessarily the same as $H^*(X)$. In general they differ by a $\lim^1$ term.
What made this spectral sequence tick is that homology/cohomology takes a cofiber sequence to a long exact sequence. When you switch from a hocolim to a holim, you are now interested in taking fiber sequences to long exact sequences of groups. Homology and cohomology don't do this, but homotopy groups do. So, there's another spectral sequence whose $E^1$-page is homotopy groups of the spaces in your diagram, which under good conditions converges to the homotopy groups of the holim. Again, one of the things that "good conditions" excludes is that you may also have $\lim^1$s pop up here.