文档介绍：page 1 of Supplement
SUPPLEMENT: THE LONG EXACT HOMOLOGY SEQUENCE AND AP-
PLICATIONS
S1. plexes
In the supplement, we will develop some of the building blocks for algebraic topology.
As we go along, we will make ments [in brackets] indicating the connection between
the algebraic machinery and the topological setting, but for best results here, please consult
a text or attend lectures on algebraic topology.
Definitions ments A plex (or simply plex) C∗ is a family
of R-,n∈ Z, along with R-homomorphisms dn : Cn → Cn−1 called differentials,
satisfying dndn+1 = 0 for all n. A plex with only finitely ’s is allowed; it
can always be extended with the aid of zero modules and zero maps. [In topology, Cn is the
abelian group of n-chains, that is, all formal binations with integer coefficients of
n-simplices in a topological space X. The map dn is the boundary operator, which assigns
to an n-simplex an n − 1-chain that represents the oriented boundary of the simplex.]
The kernel of dn is written Zn(C∗)or just Zn; elements of Zn are called cycles in
dimension n. The image of dn+1 is written Bn(C∗)or just Bn; elements of Bn are called
boundaries in dimension n. Since position of two essive differentials is 0, it
follows that Bn ⊆ Zn. The quotient Zn/Bn is written Hn(C∗)or just Hn; it is called the
nthhomology module (or homology group if the underlying ring R is Z).
[The key idea of algebraic topology is the association of an algebraic object, the col-
lection of homology groups Hn(X), to a topological space X. If two spaces X and Y are
homeomorphic, in fact if they merely have the same homotopy type, then Hn(X)and Hn(Y )
are isomorphic for all n. Thus the homology groups can be used to distinguish between topo-
logical spaces; if the homology groups differ, the spaces cannot be homeomorphic.]
Note that any exact sequence is plex, since position of essive maps is
0.
Definition A chain map f : C∗→ D∗ from a plex C∗ to a plex D∗
is a collection of