9/19/2023 0 Comments Spectral sequences![]() ![]() I do not believe their proof uses anything about the Serre spectral sequence beyond the fact it is a spectral sequence of differential graded algebras whose $E_2$ page is of the form $H^*(B K) \otimes H^*(F K)$ (when $K$ is a field), so it should give a purely algebraic result about spectral sequences with $E_2$ pages of this form. For the Leray-Serre spectral sequence, you need a fibration, and there arent that many of them around, in which you can compute the (co)homology of two out of. ![]() We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality. Roughly speaking, a spectral sequence is a system for keeping track of collections of exact sequences that have maps between them. ![]() They conclude then that transgressive elements $x_i$ of $H^*(F K)$ (generating it up to degree $N$ and such that the associated map $\Lambda \to H^*(F K)$ is injective in degree $N+1$) can be chosen such that lifts $y_i \in H^*(B K)$ of their transgressions are generators in the sense that the associated ring map $K \to H^*(B K)$ is bijective in degrees $\leq N+1$ and injective in degree $N+2$. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. In a sense there is really only one spectral sequence, just as there is only one concept of a long exact sequence (although each object may originate in a variety of settings), but there are many dierent named uses. There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realised as the limit page of a spectral. Let $B^\bullet := \bigoplus_^*(E K)$ be zero in degrees $\leq N + 2$. examples of the kinds of things spectral sequences are good for. Introduction to spectral sequences Michael Hutchings ApAbstract The words spectral sequence' strike fear into the hearts of manyhardened mathematicians. My question is about part of the Borel Transgression Theorem on spectral sequences, translated from (see also for a translated version of the whole paper): ![]()
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