The singular homology of a topological space is the simplicial homology of its singular simplicial complex:
a singular -chain on is a formal linear combination of singular simplices , and a singular -cycle is such a chain such that its oriented boundary in vanishes. Two singular chains are homologous if they differ by a boudary. The singular homology of in degree is the group of -cycles modulo modulo those that are boundaries.
Singular homology of a topological space conincide with its ordinary homology as defined more abstractly (see at generalized homology theory).
Let Top be topological space. Write sSet for its singular simplicial complex.
The groups of singular chains combine to the simplicial abelian group .
The alternating face map complex
is the singular complex of .
Its chain homology is the ordinary singular homology of .
One usually writes or just for the singular homology of in degree . See also at ordinary homology.
More generally, for any unital ring one can form the degreewise free module over . The corresponding homology is the singular homology with coefficients in , denoted .
Given a continuous map between topological spaces, and given , every singular -simplex in is sent to a singular -simplex
in . This is called the push-forward of along . Accordingly there is a push-forward map on groups of singular chains
These push-forward maps make all diagrams of the form
commute. In other words, push-forward along constitutes a chain map
It is in fact evident that push-forward yields a functor of singular simplicial complexes
From this the statement follows since is a functor.
Accordingly we have:
Sending a topological space to its singular chain complex , def. 2, and a continuous map to its push-forward chain map, prop. 1, constitutes a functor
from the category Top to the category of chain complexes.
In particular for each singular homology extends to a functor
Let be a topological space. Let be a singular 1-simplex, regarded as a 1-chain
Then its boundary is
or graphically (using notation as for orientals)
Let be a singular 2-chain. The boundary is
Hence the boundary of the boundary is
For more illustrations see for instance (Ghrist, (4.5)).
Homology of cells: disks and spheres
For all the reduced singular homology of the -sphere is
The -sphere may be realized as the pushout
which is the -ball with its boundary -sphere identified with the point. The inclusion is a “good pair” in the sense of def. 5, and so the long exact sequence from prop. 7 yields a long exact sequence
Since the disks are all contractible topological spaces we have for all by this example at reduced homology. This means that in the above long exact sequence all the morphisms
are isomorphisms, for all . Since
(by this example at reduced homology) the statement follows by induction on .
Singular homology is homotopy invariant:
A proof (via CW approximations) is spelled out for instance in (Hatcher, prop. 4.21).
Relation to homotopy groups
The singular homology groups of a topologial space serve to some extent as an approximation to the homotopy groups of that space.
For a pointed topological space, the Hurewicz homomorphism is the function
from the th homotopy group of to the th singular homology group defined by sending
a representative singular -sphere in to the push-forward along of the fundamental class .
For a topological space the Hurewicz homomorphism in degree 0 exhibits an isomorphism between the free abelian group on the set of connected components of and the degree-0 singular homlogy:
Since a homotopy group in positive degree depends on the homotopy type of the connected component of the base point, while the singular homology does not depend on a basepoint, it is interesting to compare these groups only for the case that is connected.
For a connected topological space the Hurewicz homomorphism in degree 1
is surjective. Its kernel is the commutator subgroup of . Therefore it induces an isomorphism from the abelianization :
For higher connected we have the
This is known as the Hurewicz theorem.
Relation to relative homology
For the present purpose one makes the following definition.
Write for the cokernel of the inclusion, hence for the pushout
If is a good pair, def. 5, then the singular homology of coincides with the relative homology of relative to . In particular, therefore, it fits into a long exact sequence of the form
For instance (Hatcher, theorem 2.13).
Relation to generalized homology
Singular homology computes the generalized homology with coefficients in the Eilenberg-MacLane spectrum or .
Lecture notes include
Textbook discussion in the context of homological algebra is around Application 1.1.4 of
and in the context of algebraic topology in chapter 2.1 of
and chapter 4 of
Discussion in the context of computing homotopy groups is in
Lecture notes include
Examples and applications
- Michael Barratt, John Milnor, An example of anomalous singular homology, Proceedings of the American Mathematical Society Vol. 13, No. 2 (Apr., 1962), pp. 293-297 (JSTOR)