MacLane - Homology



Springer Grundlehren Der Mathematischen Wissenschaften 114, 1963

Springer Classics in Mathematics, 1995

Homological Algebra

Algebraic Topology

"Our subject starts with homology, homomorphisms, and tensors. Homology provides an algebraic 'picture' of topological spaces, assigning to each space X a family of abelian groups $$H_0(X), \ldots, H_n(X)$$, to each continuous map $$f : X \rightarrow Y$$ a family of group homomorphisms $$f_n : H_n(X) \rightarrow H_n(Y)$$.  Properties of the space or the map can often be effectively found from properties of the groups $$H_n$$ or the homomorphisms $$f_n$$.  A similar process associates homology groups to other mathematical objects; for example, to a group $$\Pi$$ or to an associative algebra $$\Lambda$$.  Homology in all such cases is our concern."

=Chapter I : Modules, Diagrams, and Functors=

Section I.2 : Modules

 * The symbol $$\varkappa$$ is an alternate way of drawing a kappa. (It's \varkappa in TeX).

Section I.5 : Free and Projective Modules

 * The symbol $$\varrho$$ is an alternate way of drawing a rho. (It's \varrho in TeX).

Section I.8 : Functors
=Chapter II : Homology of Complexes=

Section II.1 : Differential Groups

 * Example 7: The segments q x I and p x I should be oriented upwards.

Section II.9 : Axioms for Homology
=Chapter III : Extensions and Resolutions=

=Chapter IV : Cohomology of Groups=

Section IV.1 : The Group Ring

 * this means more exactly that $$\mu_0 y$$ is that function on $$\Pi$$ to Z for which... : i.e., if we regard elements of $$\Pi(Z)$$ as functions $$\Pi \rightarrow Z$$ sending $$x \mapsto m(x)$$.

Section IV.2 : Crossed Homomorphisms

 * The motivation for studying $$f_a = xa - a$$ is to look at fixed points of modules.

=Chapter V : Tensor and Torsion Products=

=Chapter VI : Types of Algebras=

=Chapter VII : Dimension=

=Chapter VIII : Products=

=Chapter IX : Relative Homological Algebra=

=Chapter X : Cohomology of Algebraic Systems=

=Chapter XI : Spectral Sequences=

=Chapter XII : Derived Functors=