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This document explains spectral sequences and homology in a concise and clear way. It delves into the basics of chain complexes, differentials, and the important notation of spectral sequences. The focus is placed on the Serre spectral sequence and its stabilization.

The homology definition$H_n(A_\bullet)$ of a chain complex is defined for arbitrary homology theories as the quotient$Z_n / B_n$. With respect to spectral sequences, there exists in a chain complex a differential that goes out of the group, in this case$d_n$, and one that goes into the group, in this case$d_{n+1}$. This leads to a powerful framework for examining various homology groups depending on which differentials we use. These considerations are organized into what are calledpages.

A spectral sequence behaves like a book with an infinite number of pages. Each page represents a two-dimensional grid of groups assigned to specific differentials. We move from one page to the next using a natural transformation, and ideally, the pages stabilize to apage limit in the infinite.

The notation for homology groups is$E^r_{p,q}$, where$r$ denotes the page number from a totally ordered index set$(I,\leq)$, and$p, q \in \mathbb{Z}$ are the horizontal and vertical indices.

The differentials on each page$r$ are heavily dependent on the definition of the spectral sequence. We define the differentials$d_{IN}$ and$d_{OUT}$ as the incoming and outgoing differentials of$E^r_{p,q}$, and the operation from one page to the next is:$E^{r+1}_{p,q}$ which is defined as de kernel of the outgoing map modulo the image of the incoming map.

The Serre spectral sequence is a concrete example where the sequence stabilizes. There exists an$R$ such that for all$r > R$, we have:$E^r_{p,q} = E^R_{p,q}$. These entries form the stabilized page$E^\infty_{p,q}$, and we will go over the essential concepts needed to populate the spectral sequence.

For a printable version of this material in PDF format,click here to download.

• Spectral Sequences: A versatile tool for homological computations.
• Stabilization: Key insight of the Serre spectral sequence.
• Visual Representation: Pages behave like two-dimensional chain complexes.

For deeper exploration into homology theories and spectral sequences, here are some resources:

• Homology and Spectral Sequences – MathWorld
• Introduction to Spectral Sequences – SAGE

Enjoy your journey into the fascinating world of algebraic topology and homology theory!