A Concise Introduction to Spectral Sequences
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Contents
- 1 Basics of Spectral Sequences
- 1.1 Definitions
- 1.2 Filtered Complexes
- 1.3 Double Complexes
- 1.4 Exact couples
- 1.5 Calculations
- 2 Applications in Topology
- 2.1 Mini-dictionary of Topology
- 2.2 The Leray-Serre Spectral sequence
- 2.3 The Eilenburg-Moore Spectral sequence
- 2.4 The Bockstein Spectral Sequences
- 3 Applications in Algebra
- 3.1 Mini-dictionary of Algebra
- 3.2 The Knneth Spectral Sequences
- 3.3 The Hoshchilds-Serre Spectral Sequences
- 3.4 The Grothendieck Spectral Sequences
- A Cohomology for Topological Groups
- A.1 Comparison Theorem
- A.2 Transgression
- A.3 The Borel Theorem
- A.4 Cohomology Computation
- B Cohomology for Compact Lie Groups
- B.1 The Koszul Complex
- B.2 Some Commutative Algebra
- B.3 Flag Manifolds
- B.4 Cohomology Computation
- C Cohomology for Discrete Groups
- C.1 Equivariant cohomology
- C.2 The Cartan–Leray Spectral Sequence
Preface
In this book, I would like to give an acceptable, clear, and concise introduction to spectral sequences. The preliminary is basic homological algebra and basic algebraic topology. Two mini-dictionaries are included for last two chapters.
The first chapter is the most original part. It contains the short proof by the author, and with detailed check. It is not painful and only the elementary stuff are left to reader.
The second chapter is about topology. I introduced the spectral sequences which are easy to introduce. It is a pity that I do not mention enough examples in topology. The interested reader could read [1] and [3] for examples and deeper topics.
The last chapter is about algebra. Here, we used only the spectral sequences for double complex. They are almost all the spectral sequence as I know in algebra which can be introduced shortly. The curious reader is encouraged to ask [3] for more examples.
Last but no mean least, enjoy spectral sequences !
Title
Dedication
