## FANDOM

36 Pages

"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 FunctorsEdit

## Section I.2 : ModulesEdit

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

## Section I.5 : Free and Projective ModulesEdit

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

# Chapter II : Homology of ComplexesEdit

## Section II.1 : Differential GroupsEdit

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

# Chapter IV : Cohomology of GroupsEdit

## Section IV.1 : The Group RingEdit

• 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 HomomorphismsEdit

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