I am trying to compute $H^*(X)$ for a (potentially large, finite, finitely filtered) simplicial complex $X$ using a cover $U_i$ of $X$.
I am building chain complexes for $X$ with a simplex that appears in the $p$th stage of the complex filtration generating a basis element of $C_*X$ with degree $p$. In cochains, this means that a simplex $\sigma$ that appears in the $p$th stage of a complex with $N$ filtration steps generates a relation $t^{N-p}\sigma=0$ in the cochain complex.
I want to use the generalized Mayer-Vietoris spectral sequence: $E_0$ is a double complex where the $p$-th column is $\oplus_{J\subset [n], |J|=p+1} C^*(\cap_{i\in J}U_i)$ with the usual coboundary, and the row coboundary is the restriction maps with signs from the nerve complex of the covering. In particular, I want the version of the spectral sequence of a double complex that starts with vertical cohomology and goes from there.
I already know how to compute the spectral sequence itself (even with algorithms!) -- my problem is with inferring $H^*(X)$ from the $E_\infty$-page. I know that $E_\infty=\mathrm{Gr} H^*(\mathrm{Tot}(E_0))$ and that $H^*(\mathrm{Tot}(E_0)) = H^*(X)$. Furthermore, I know that there are maps $H^n(X)\to E_\infty^{0,n}$ and $E_\infty^{n,0}\to H^n(X)$.
My questions are:
- Can I always find a representative in $\mathrm{Gr}H^*(\mathrm{Tot}(E_0))$ of any element of $E_\infty^{p,q}$ that has the form $(a_0,\dots,a_p,0,\dots,0)$ with all the $a_i$ nonzero (on a chain level)? How?
- Can I always find a representative in $\mathrm{Gr}H^*(\mathrm{Tot}(E_0))$ of any element of $E_\infty^{p,q}$ that has the form $(0,\dots,0,a_p,a_{p+1},\dots,a_n)$ with all the $a_i$ nonzero (on a chain level)? How?
- Can I always find a representative in $\mathrm{Gr}H^*(\mathrm{Tot}(E_0))$ of any element of $E_\infty^{p,q}$ that has the form $(a_0,\dots,a_{p-1},a_p,a_{p+1},\dots,a_n)$ with all the $a_i$ nonzero (on a chain level)? How?
- Is there a reasonably easy way to concretely write down the map $E_\infty^{n,0}\to H^*(X)$? In particular, if I have a $0$-cocycle in a $k$-fold intersection of cover elements, can I then write down a $k$-cocycle in $X$ directly from that $0$-cocycle? How?