- Relative homology
In
algebraic topology , a branch ofmathematics , the (singular) homology of a topological space relative to a subspace is a construction insingular homology , forpairs of spaces . The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolutehomology group comes from which subspace.Definition
Given a subspace Asubset X, one may form the
short exact sequence :0 o C_ullet(A) o C_ullet(X) o C_ullet(X) /C_ullet(A) o 0
where C_ullet(X) denotes the
singular chain s on the space "X". The boundary map on C_ullet(X) leaves C_ullet(A) invariant and therefore descends to a boundary map on the quotient. The correponding homology is called relative homology::H_n(X,A) = H_n (C_ullet(X) /C_ullet(A)).
One says that relative homology is given by the relative cycles, chains whose boundaries are chains on "A", modulo the relative boundaries (chains that are homologous to a chain on "A", i.e. chains that would be boundaries, modulo "A" again).
Properties
The above short exact sequences specifying the relative chain groups gives rise to a chain complex of short exact sequences. An application of the
snake lemma then yields along exact sequence :cdots o H_n(A) o H_n(X) o H_n (X,A) stackrel{delta}{ o} H_{n-1}(A) o cdots .
The connecting map "δ" takes a relative cycle, representing a homology class in "Hn"("X", "A"), to its boundary (which is a chain in "A").
It follows that "Hn"("X", "x"0), where "x"0 is a point in "X", is the "n"-th
reduced homology group of "X". In other words, "Hi"("X", "x"0) = "Hi"("X") for all "i" > 0. When "i" = 0, "H"0("X", "x"0) is the free module of one rank less than "H"0("X"). The connected component containing "x"0 becomes trivial in relative homoloy.The
excision theorem says that removing a sufficiently nice subset "Z" ⊂ "A" leaves the relative homology groups "Hn"("X", "A") unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that "Hn"("X", "A") is the same as the "n"-th reduced homology groups of the quotient space "X"/"A".The "n"-th local homology group of a space "X" at a point "x"0 is defined to be "Hn"("X", "X" - "x"0). Informally, this is the "local" homology of "X" close to "x"0.
Relative homology readily extends to the triple ("X", "Y", "Z") for "Z" ⊂ "Y" ⊂ "X".
One can define the
Euler characteristic for a pair "Y" ⊂ "X" by:chi (X, Y) = sum _0 ^n (-1)^j ; mbox{rank} ; H_j (X, Y).
The exactness of the sequence implies that the Euler characteristic is "additive", i.e. if "Z" ⊂ "Y" ⊂ "X", one has
:chi (X, Z) = chi (X, Y) + chi (Y, Z).,
Functoriality
The map C_ullet can be considered to be a
functor :C_ullet:old{Top}^2 oold{Comp}
where Top2 is the
category of pairs of topological spaces and Comp is thecategory of chain complexes .References
*
* Joseph J. Rotman, "An Introduction to Algebraic Topology", Springer-Verlag, ISBN 0-387-96678-1ee also
*
Relative contact homology
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