Book contents
- Frontmatter
- Contents
- General Introduction
- PART I HOMOLOGY THEORY OF POLYHEDRA
- Background to Part I
- 1 The Topology of Polyhedra
- 2 Homology Theory of a Simplicial Complex
- 3 Chain Complexes
- 4 The Contrahomology Ring for Polyhedra
- 5 Abelian Groups and Homological Algebra
- 6 The Fundamental Group and Covering Spaces
- PART II GENERAL HOMOLOGY THEORY
- Bibliography
- Index
3 - Chain Complexes
from PART I - HOMOLOGY THEORY OF POLYHEDRA
Published online by Cambridge University Press: 02 February 2010
- Frontmatter
- Contents
- General Introduction
- PART I HOMOLOGY THEORY OF POLYHEDRA
- Background to Part I
- 1 The Topology of Polyhedra
- 2 Homology Theory of a Simplicial Complex
- 3 Chain Complexes
- 4 The Contrahomology Ring for Polyhedra
- 5 Abelian Groups and Homological Algebra
- 6 The Fundamental Group and Covering Spaces
- PART II GENERAL HOMOLOGY THEORY
- Bibliography
- Index
Summary
Chain and contrachain complexes
An oriented simplicial complex ‡ determines, for each dimension p, a chain group Cp and a boundary homomorphism ∂: Cp → Cp − 1 From these data the homology and contrahomology groups may be obtained. We now propose to confine attention to these purely algebraical concepts and accordingly define
3.1.1 Definition A chain complex C. = {Cp, ∂p}, is a collection of abelian groups Cp, one for each integer p, and of (right) homomorphisms ∂p : Cp → Cp − 1 such that ∂p ∂p − 1 = 0, for each p.
Dually, we define
3.1.1c Definition A contrachain complex, C = {Cp, δp} is a collection of abelian groups Cp, one for each integer p, and of (left) homomorphisms δp : Cp → Cp + 1 such that δp + 1δp = 0, for each p.
We shall generally write C for C, and shall often write ∂, δ for ∂p, δp.
It is clear how we may define the homology groups Hp (C) of the chain complex C; if Zp or Zp (C), the p-th cycle group, is the kernel of ∂p and Bp or Bp (C), the p-th boundary group, is the image of ∂p + 1, then Bp is a subgroup of the abelian group Zp and Hp (C) is the factor group Zp\Bp. The definition of Hp (C) is analogous; if the context makes it clear that a contrachain complex is in question, we may suppress the superscript dot, so that we may write Hp (C).
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- Homology TheoryAn Introduction to Algebraic Topology, pp. 95 - 139Publisher: Cambridge University PressPrint publication year: 1960