If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal coefficient theorem for cohomology.
Now, suppose I have a fibration $F \to E \to B$ such that $F$ and $B$ have homology and cohomology over $k$ of finite type in each degree and $\pi_1(B) = 0$ for simplicity. Certainly, the $E_2$-page of the cohomology Serre spectral sequence will be dual to the $E^2$-page homology Serre spectral sequence. My first question is: Is it also true that the differentials for the cohomology spectral sequence are dual as a linear map to the differentials of the homology spectral sequence, and vice versa?
Secondly, the cohomology Serre spectral sequence is a multiplicative one. Is the homology one comultiplicative? If so, is the product for cohomology dual to the coproduct for homology?
Finally, if all of this holds, to which extend can it generalized?
Best Answer
Yes, this is the case. This is easiest to see using the exact couple formalism. Suppose you have an exact couple, meaning a long exact sequence consisting of maps $i: D \to D$, $j: D \to E$, $k: E \to D$, where all the terms are (possibly graded) vector spaces over a field. Because dualization is exact (as is taking levelwise duals of graded objects), you can verify that the sequence of maps $i^*: D^* \to D^*$, $k^*: D^* \to E^*$, $j^*:E^* \to D^*$ also forms an exact couple and the associated spectral sequence is dual to the original spectral sequence.
The "short" construction of the Serre spectral sequence is obtained in the case where B is a CW-complex (one can reduce to this case by replacing B with a weakly equivalent object), and filtering the total space by the preimages of the skeleta. The long exact sequences in homology and cohomology associated to this filtration are dual to each other, and so when one creates the graded vector spaces forming the exact couple by summing up one gets that the homology and cohomology exact couples are dual to each other.