Effective Homology and Spectral Sequences 1st Edition by Ana Romero Ibanez ISBN 8469103423 9788469103425

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Product details:

ISBN 10:8469103423 
ISBN 13 :9788469103425
Author:Ana Romero Ibanez

It seems that after several years devoted to this work, with a lot of satisfactions and
a few disappointments, all the effort is worth it and this memoir is coming to an end. It
has been a long way, especially hard over the last months, and it would not have been
possible without the help and the support of many people who have been close to me.
Although it is difficult to name them all, I would like to express my gratitude to, at
least, a few of them.

 Table of contents:

Preliminaries
Basics on Homological Algebra
Chain complexes
Spectral sequences
Simplicial Topology
Simplicial sets
Homotopy groups
Eilenberg-MacLane spaces
Effective Homology
Definitions and fundamental results
Perturbation theorems
The Kenzo program

Effective homology and spectral sequences of filtered complexes: algorithms and programs
Filtrations and spectral sequences
Main theoretical results
An algorithm computing spectral sequences of filtered complexes
Effective chain complexes
Groups
Locally effective chain complexes
Bicomplexes
Effective homology of a bicomplex
Spectral sequence of a bicomplex
Examples
A didactic example
The Bar construction
Implementation
A new module for the Kenzo system
Examples
Bicomplexes
The Bar construction

Effective homology and spectral sequences of filtered complexes: applications
Serre spectral sequence
Introduction
Effective homology of a twisted product
An algorithm computing the Serre spectral sequence
Implementation and examples
K(Z, 1) ×τ S²
Postnikov tower
Eilenberg-Moore spectral sequence
Introduction
Effective homology of the fiber space of a fibration
An algorithm computing the Eilenberg-Moore spectral sequence
Loop spaces

Effective homology of free simplicial Abelian groups
Previous definitions and results
The Dold-Kan correspondence
Some remarks about Eilenberg-MacLane spaces
Effective homology of Eilenberg-MacLane spaces
Another model for Eilenberg-MacLane spaces
The free simplicial Abelian group RX
Effective homology of RX
Left equivalence
Right equivalence
Final result
Implementation
The simplicial Abelian group RX
Effective homology of RX

Effective homology and Bousfield-Kan spectral sequence
Some algorithms for cosimplicial structures
Cosimplicial objects
Cosimplicial Abelian groups
Cosimplicial simplicial Abelian groups
Definitions and fundamental results
Homotopy spectral sequence of a cosimplicial simplicial Abelian group
Cosimplicial simplicial sets
Definitions and examples
Homotopy spectral sequence of a cosimplicial space
Construction of the Bousfield-Kan spectral sequence
Study of the first level of the spectral sequence
Algorithms computing E₁ and E₂
Toward an algorithm computing the Bousfield-Kan spectral sequence
Homotopy spectral sequence of a tower of fibrations
Another definition for the homotopy spectral sequence of a cosimplicial space
Sketch of the algorithm

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