In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the (singular) homology of a topological space relative to a subspace is a construction in
singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
, for
pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
comes from which subspace.
Definition
Given a subspace
, one may form the
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
:
where
denotes the
singular chain
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
s on the space ''X''. The boundary map on
descends to
and therefore induces a boundary map
on the quotient. If we denote this quotient by
, we then have a complex
:
By definition, the
th relative homology group of the pair of spaces
is
:
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
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance ...
then yields a
long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
:
The connecting map ''
'' takes a relative cycle, representing a homology class in
, to its boundary (which is a cycle in ''A'').
It follows that
, where
is a point in ''X'', is the ''n''-th
reduced homology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise stat ...
group of ''X''. In other words,
for all
. When
,
is the free module of one rank less than
. The connected component containing
becomes trivial in relative homology.
The
excision theorem
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A, the theorem ...
says that removing a sufficiently nice subset
leaves the relative homology groups
unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that
is the same as the ''n''-th reduced homology groups of the quotient space
.
Relative homology readily extends to the triple
for
.
One can define the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
for a pair
by
:
The exactness of the sequence implies that the Euler characteristic is ''additive'', i.e., if
, one has
:
Local homology
The
-th local homology group of a space
at a point
, denoted
:
is defined to be the relative homology group
. Informally, this is the "local" homology of
close to
.
Local homology of the cone CX at the origin
One easy example of local homology is calculating the local homology of the
cone (topology)
In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching ''X'' into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by CX or by \operatorname(X).
...
of a space at the origin of the cone. Recall that the cone is defined as the quotient space
:
where
has the subspace topology. Then, the origin
is the equivalence class of points