# Serre Spectral Sequence. Motivation

• WWGD
In summary, the conversation discusses Serre's spectral sequence and how it relates to fibrations. The process involves defining a filtered chain complex with a differential, and then following a recipe for constructing a spectral sequence. The general ingredients for a spectral sequence include starting with a sheet E_ro and then moving to the next sheet, which is isomorphic to the homology of E_ro. The conversation also requests comments on the Serre spectral sequence and spectral sequences in general.
WWGD
Gold Member
Hi, everyone. I don't know if this is the right forum, here it goes:

I have been reading on Serre's spectral sequence (Wiki, Hatcher). I understand the general goal, but I don't get well the process:

how does a fibration F->E->B

give rise to a spectral sequence?.

Thanks.

I guess I should be less impulsive in my posting: I just remembered that the base

B is assumed a CW-complex, and that we consider the fiber of the n-skeleton,

and this fiber somehow has the structure of a filtered chain complex. we somehow

define in this filtered complex, a differential. Then we just have a filtered chain complex,

and we follow the recipe for it .Still, the details are fuzzy.

Also: what are the general ingredients for a spectral sequence?. We start with

a sheet E_ro (ro=0,1,2, usually, ABAIK---Asbestos I know), then the next sheet

is isomorphic to the homology of E_ro, and we then go on, until the sheets converge

(we assume the sheets are non-zero only in the first quadrant, so that we run out

of groups as we approach the quadrants.)

Anyway, I would appreciate some comments, both on the Serre spectral, and

in general for spectral sequences.

Thanks.

## What is the Serre Spectral Sequence?

The Serre Spectral Sequence is a mathematical tool used in algebraic topology to compute the homology and cohomology of spaces that are constructed by "gluing" simpler spaces together.

## What is the motivation for using the Serre Spectral Sequence?

The main motivation for using the Serre Spectral Sequence is to break down the computation of a complicated space into smaller, more manageable pieces. This allows for easier computations and better understanding of the topology of the space.

## How is the Serre Spectral Sequence constructed?

The Serre Spectral Sequence is constructed using the concept of fibrations, which are continuous maps that "look like" the projection map of a product space. It is then derived from the long exact sequence of homotopy groups associated with a fibration.

## What are some applications of the Serre Spectral Sequence?

The Serre Spectral Sequence has many applications in algebraic topology, including calculating the homology and cohomology of spaces, computing the fundamental group of a space, and studying the homotopy type of a space.

## Are there any limitations to the Serre Spectral Sequence?

The Serre Spectral Sequence can only be used for spaces that can be constructed by "gluing" simpler spaces together, such as manifolds and CW complexes. It also requires a significant level of mathematical understanding to use effectively.

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